The convex hull property and topology of hypersurfaces with nonnegative curvature

http://www.elsevier.com/locate/aim

Advances in Mathematics 180 (2003) 324–354

.

The convex hull property and topology ofhypersurfaces with nonnegative curvature

Stephanie Alexandera and Mohammad Ghomib,1,*aDepartment of Mathematics, University of Illinois, Urbana, IL 61801, USA

bDepartment of Mathematics, University of South Carolina, Columbia, SC 29208, USA

Received 15 May 2001; accepted 5 August 2002

Communicated by Michael J. Hopkins

Abstract

We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed

hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies

certain required conditions. This gives a convex hull property, dual to the classical one for

surfaces with nonpositive curvature. A version of this result in the nonsmooth category is

obtained as well. We show that our boundary conditions determine the topology of the surface

up to at most two choices. The proof is based on uniform estimates for radii of convexity of

these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing

procedure.

r 2003 Elsevier Science (USA). All rights reserved.

MSC: primary 53A07; 53C45; secondary 53C21; 53C23

Keywords: Convex hull; Nonnegative curvature; Convergence theorem; Locally convex surface; Convex

cap; Clipping

1. Introduction

The main objects of study in this paper are locally convex hypersurfaces withboundary immersed in Euclidean space, and their infinitesimal counterparts, smooth

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*Corresponding author. Fax: +1-831-459-8360.

E-mail addresses: [email protected] (S. Alexander), [email protected] (M. Ghomi).

URLs: http://www.math.uiuc.edu/~sba, http://www.math.sc.edu/~ghomi.1Partially supported by NSF Grant DMS-0204190.

0001-8708/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0001-8708(03)00006-9

hypersurfaces of nonnegative sectional curvature. In contrast to the complete casewithout boundary, which has been well studied [17,29,18,19], the behavior of theseobjects is in general quite varied and complex [12]. Thus, it is desirable to find naturalboundary conditions which induce nice global behavior. Our primary motivation inthis study is the classical convex hull property [26] which states that in Euclideanspace a compact nonpositively curved hypersurface lies within the convex hull of itsboundary. We show that this phenomenon has a dual, and in the process developsome basic tools for studying locally convex immersions. Our principal result in thesmooth category is as follows:

Theorem 1.1. Let M be a compact, connected, and smooth n-manifold, nX2;

f : M-Rnþ1 be a smooth immersion with nonnegative sectional curvature, and C :¼conv f ð@MÞ denote the convex hull of the image of the boundary of M (where we

assume @Ma|). Suppose that:

(1) f ð@MÞC@C;(2) f has positive curvature on @M;(3) f is an embedding on each component of @M:

Then the image of the interior of M lies completely outside the convex hull of the image

of its boundary:

f ðint MÞ-C ¼ |:

Further, f is an embedding on @M; and M is homeomorphic to the closure of a

component of @C � f ð@MÞ with boundary f ð@MÞ: (If C is degenerate, i.e. int C ¼ |;then we take @C to be two copies of C glued along their relative boundaries.)

It is obvious that, as far as our convex hull property (CHP) is concerned,condition (1) in Theorem 1.1 is necessary. Conditions (2) and (3) cannot be omitted,as demonstrated in Figs. 1(a) and (b), respectively. Also note that even thoughTheorem 1.1 implies that f ð@MÞ is embedded, in general f ðMÞ is not embedded, seeFig. 1(c). Theorem 1.1 implies that the topology of M is uniquely determined by the

image of @M if @M has multiple components or is homeomorphic to Sn�1: But if @M

is a torus and f : @M-@C is a knotted embedding, then the components of

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(a) (b) (c)

Fig. 1.

S. Alexander, M. Ghomi / Advances in Mathematics 180 (2003) 324–354 325

@C � f ð@MÞ are not homeomorphic [13], and so in general the topology of M isdetermined only up to two choices. This degree of topological rigidity is significant,since it fails if condition 1 of Theorem 1.1 is violated: Fig. 2 illustrates how toconstruct a curve which, for any given k; bounds k topologically distinct embeddedsurfaces of positive curvature.

The first step of the proof of Theorem 1.1 (Section 3) is to show that f is locally

convex and one-sided, as defined below. This involves extending certain aspects ofprevious work of Sacksteder [29], and Greene and Wu [14], and is the only placewhere smoothness is used in the proof of the Theorem 1.1. This theorem then followsfrom its counterpart in the nonsmooth setting, Theorem 1.2.

We say the immersion f : M-Rnþ1 is locally convex if f has an extension f to a

manifold without boundary M; where every point pAM has a neighborhood Up in

M that is embedded by f into the boundary of a convex body KpCRnþ1: (The

assumption that f extends to a collaring manifold M is a compatibility condition onthe Kp for boundary points p; see the discussion below.) f is one-sided if Up and Kp

may be chosen so that the side of f ðUpÞ on which Kp lies varies continuously with p:

(Fig. 1(a) illustrates an immersion which is locally convex but not one-sided.) If, forall pA@M; Up may be chosen so that f ðUp-MÞ contains no line segments, we say f

is locally strictly convex on a neighborhood of @M:

Theorem 1.2. Let M be a compact connected topological n-manifold, nX2;

f : M-Rnþ1 be a locally convex immersion, and C :¼ conv f ð@MÞa|: Suppose that:

(1) f ð@MÞC@C;(2) f is locally strictly convex on a neighborhood in M of @M;(3) f is an embedding on each component of @M:

Then f satisfies the strong CHP:

f ðint MÞ-C ¼ |:

Further, f is an embedding on @M; and M is homeomorphic to the closure of a

component of @C � f ð@MÞ with boundary f ð@MÞ: If condition (2) is replaced by

ð20Þ f is one-sided, then the same conclusions hold, but with the strong CHP replaced

by the CHP:

f ðMÞ-int C ¼ |:

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Fig. 2.

S. Alexander, M. Ghomi / Advances in Mathematics 180 (2003) 324–354326

There is a nonintuitive distinction between our notion of local convexity and whatmay be referred to as weak local convexity. By the latter we mean that for each pointpAM the restriction of f to some neighborhood of p in M extends to an embedding ofa manifold without boundary into the boundary of a convex body Kp: Thus weak

local convexity does not require that the local extensions across the boundary maybe made compatible. The two notions of local convexity are equivalent in the settingof Theorem 1.1 (see Lemma 3.2). On the other hand, Fig. 3 illustrates a weaklylocally convex immersion (a box with a hole and a folded-down tab that pierces thebox) for which the CHP fails (since the box clearly intersects the convex hull of itsboundary). This example satisfies (1) and (3) of Theorem 1.2, and is one-sided in thesense that it is one sided on the interior of M and locally has extensions across theboundary that are convex on the same side. In short, Theorem 1.2 is false for one-sided weakly locally convex immersions.

A special case of Theorem 1.2 where stronger conclusions are possible occurswhen each component of @M lies in some hyperplane. We give a comprehensivetreatment of this case in Section 5, where our result complements work of Rodriguez[28, Theorem 4] and Kuhnel [21] on tight immersions of manifolds with boundary.

The basic strategy for proving Theorem 1.2 is highly intuitive, and roughly may bedescribed as follows. For simplicity, we suppress the immersion f : We wish to showthat if @M lies on the boundary of its own convex hull C; then all points of M lieoutside int C: If points in M lie outside C; we ‘‘clip’’ them off with planes in such away that C remains intact; replacing the clipped-off part of M with a topologicaldisk in the clipping plane, we obtain a new locally convex surface with the sametopology and boundary (Section 4). We continue to clip and modify M until in thelimit, there are no points outside C: We then have a locally convex surface M 0 withthe same boundary as M and all points inside C: We extend this surface to a closedlocally convex surface without boundary by gluing on portions of @C (Section 8). Bya theorem of van Heijenoort (Lemma 4.1), the resulting surface must lie on theboundary of a convex body. That convex body obviously contains C and, hence, M 0

lies in @C: Since we did not clip any points in M that were also in int C; there mustnot have been any.

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Fig. 3.


One of the principal challenges in this proof is controlling the limit of theclippings. To this end, we find uniform bounds for radii of convexity (Section 6).These bounds may be considered as generalized a priori curvature estimates, andlead to an analogue of Blaschke’s selection principle for surfaces with boundary(Section 7). The remaining key ingredient is the somewhat subtle gluing argument(Section 8).

A version of the convex hull properties proved in this paper was originallyconjectured in the second author’s Ph.D. thesis [9, Conjecture E.0.5]. A proof of aversion of Theorem 1.1, assuming local convexity, was obtained subsequently to thepresent work by Guan and Spruck [16], using their work on Monge-Ampereequations and our Theorem 7.1. Labourie has investigated the singularities in theconvergence of smooth locally convex hypersurfaces in Euclidean space underuniform upper bounds on sectional curvature [22]. For inequalities relating intrinsicand extrinsic invariants of locally convex hypersurfaces, see [23] and Gromov [15, p.280]. Ends of complete, smooth locally convex hypersurfaces with compactboundary were studied by Currier and the first author [1]. For an application ofour main results, see [10, Appendix A], where it is proved that there exists a smoothsimple closed curve without inflection points which lies on the boundary of a convexbody, but bounds no surfaces of positive curvature. This result, which is related to aquestion of Yau [31, Problem 26] on characterizing curves which bound surfaces ofpositive curvature, was one of the original motivations for developing the convexhull properties proved in this paper. Finally, we should mention that a version ofCHP for noncompact surfaces will appear in [32].

2. Basic definitions and notation

By M; we always mean a connected n-dimensional manifold with boundary, wherenX2: The boundary and interior of M are denoted, respectively, by @M and int M :¼ M � @M: By smooth we mean infinitely differentiable ðCNÞ: But with theexception of Section 3, and unless stated otherwise, the manifolds and mappings inthis paper are not a priori assumed to have any degree of regularity. By an immersion

we mean a locally one-to-one continuous map. An embedding is a mapping which is a

homeomorphism onto its image. We say f : M-Rnþ1 is a smooth immersion if M

and f are smooth, and f has nonvanishing Jacobian.

The standard Euclidean innerproduct is denoted by /�; �S; and jj � jj :¼ /�; �S1=2 is

the corresponding norm. Sn and Bnþ1 denote, respectively, the unit sphere and closed

unit ball in Rnþ1: For any pair of subsets X ; YCRnþ1 we have the Euclidean distance:

distðX ;YÞ :¼ inff jjx � yjj j xAX and yAYg

and the Hausdorff distance:

distHðX ;YÞ :¼ inffrX0 j XCY þ rBnþ1 and YCX þ rBnþ1g:

ARTICLE IN PRESSS. Alexander, M. Ghomi / Advances in Mathematics 180 (2003) 324–354328

By a convex body in Rnþ1 we mean a compact convex set with nonempty interior. For

any subset XCRnþ1; int X and conv X denote, respectively, the interior and convex

hull of X : An intersection of finitely many closed halfspaces is called a convexpolyhedron.

Recall that by saying f : M-Rnþ1 is locally convex, we mean f has an extension f

to a manifold without boundary M; where every point pAM has a neighborhood Up

in M that is embedded by f into the boundary of a convex body Kp: If, for all pA@M;

Up may be chosen so that f ðUp-MÞ contains no line segments, we say f is locally

strictly convex on a neighborhood of @M: (It is not ruled out that fðUpÞ may contain

line segments.) If we may choose Up ¼ M; we say f is a convex embedding. If the side

of f ðUpÞ on which Kp lies, which we call the positive side, varies continuously with p;

we say that f is one-sidedly locally convex, or merely one-sided.

If f is locally convex, then each neighborhood Up may be chosen so that fðUpÞ issupported by a hyperplane Hp at f ðpÞ; and is the graph of a convex, Lipschitz height

function over an open disk in Hp [5]. Since M is connected, it easily follows that any

two points of M can be joined by an arc whose image under f has finite length.

Therefore f induces an intrinsic distance on M; which we denote by distf ðp; qÞ: Thisdistance is locally bi-Lipschitz equivalent to the Euclidean distance [5, pp. 6, 79].Since each neighborhood Up is bi-Lipschitz equivalent to a Euclidean disk, we obtain

an atlas on M for which the transition functions are bi-Lipschitz. When we refer to a

Lipschitz structure on M; this is the one we mean.Note that we have made no regularity assumptions on @M; which might, for

example, exhibit fractal behavior. Thus there may be no arc in M; from a givenboundary point to another point of M; whose image has finite length. Therefore theintrinsic distance distf ; while always finite on pairs of points in int M;may take value

N on pairs that include a point of @M: In any case, completeness of M under distf is

defined as usual, namely, Cauchy sequences in M converge in M:

3. Local convexity and one-sidedness

In this section, we show that if an immersion satisfies the conditions of Theorem1.1, then it is one-sidedly locally convex. We also examine one-sidedness innonsmooth locally convex immersions.

First, we give a condition guaranteeing that any component of the set of flat points

of a smooth nonnegatively curved hypersurface is a convex set. By flat points wemean the set where the second fundamental form vanishes. Our Lemma 3.1generalizes both Theorem 1 of [29], where M is assumed to be complete and withoutboundary, and a proposition in [14], where M is assumed to be homeomorphic to the2-sphere with finitely many points deleted. Both our proof and that of [14] use themethod of [29], to which we refer for certain arguments; however, our proof isshorter and more direct than the other two.

ARTICLE IN PRESSS. Alexander, M. Ghomi / Advances in Mathematics 180 (2003) 324–354 329

Below, by the completion of M; we mean its completion in the intrinsic metricdistf : The Cauchy boundary consists of the added points.

Lemma 3.1. Let f : M-Rnþ1 be a smooth immersion with nonnegative curvature,where M is a manifold without boundary. Suppose f has strictly positive curvature on

M-U ; where U is a neighborhood of the Cauchy boundary of M in the completion of

M: Then the restriction of f to each component of the set of flat points of f is an

embedding onto a convex set.

Proof. Let X be a component of the set of flat points of f : Since the Gauss map hasrank 0 on X ; Sard’s theorem [8, 3.4.3] implies its image has vanishing one-dimensional Hausdorff measure, hence is totally disconnected (here we are using thefact that f is CN). Therefore the Gauss map is constant on X ; say n0: Let h : M-R

be the height function given by hðpÞ :¼ /f ðpÞ; n0S: Then the differential of h vanisheson X : Consequently, by a theorem of Morse [24], f ðX Þ lies in a hyperplane H

orthogonal to n0:Choose a point p in X ; and let S be the maximal subset of X that is starshaped

about p, that is, S consists of all points that can be joined to p by an arc g in X that is

mapped homeomorphically by f onto a line segment gn in H (we call g an H-segment). It suffices to show S ¼ X : Indeed, since p is arbitrary, it follows from thisthat f ðXÞ is a convex subset of H and the restriction of f to X is an isometry.

By assumption, all Cauchy sequences in X ; with respect to the metric distf on M;

converge in X : Therefore the closed distf -balls in X are compact. It follows then by

an Arzela–Ascoli argument [4, (5.16)] that S is closed in X : Thus to complete theproof it remains to show that S is open in X :

Suppose S is not open in X : Then there is a point qAS and a sequence qiAX � S

converging to q: If g is an H-segment in X from p to q; then g has a tubularneighborhood W in M that is homeomorphic to an open disk. Since f is tangent toH along g; we may choose W so small that f ðWÞ is the homeomorphic image of W

and is the graph of a smooth function g:W n-R; where W n is the homeomorphic

image of f ðWÞ under orthogonal projection to H: Let gni be the segment in W n from

f ðpÞ to f ðqiÞ; and gi be the arc in W whose f -image projects orthogonally to gni : SinceqieS; then gigX ; and so gi intersects some component A of the set of nonflat points

of W : Equivalently, gni intersects the orthogonal projection An of f ðAÞ to H: Note

that An is the nonvanishing set of the hessian of g:Now we obtain a contradiction by showing that gni cannot intersect the orthogonal

projection An: In a simply connected manifold, any component of the complement ofan open connected set has connected boundary [29, Corollary 1]. Therefore, theboundary in W of each component U of W � A is connected. The boundary of U inW ; being a connected set of flat points, lies in a hyperplane (by the argument above),and the normal to the hyperplane agrees with the normal to f on the boundary of U :

But then there is a unique C1 function gA on W n that agrees with g on An and agrees

with some linear function on each component of W n � An: In fact, since the hessian

of g vanishes on the boundary of An in W n; it follows that g and gA agree up to third


order on the boundary of An; and hence gA is C2: Since the nonvanishing set An ofthe hessian of gA is connected, we may assume that the hessian of gA is positive

semidefinite, and hence that gA is convex on W n: Since gA; its differential and itshessian all vanish at f ðpÞ and f ðqiÞ; the convexity of gA implies that they vanish at

every point of gni (this claim is easily verified geometrically, or by calculation as in

[29, p. 616] or [14, p. 462]). Since gA and g agree on the open set An; and the hessian

of g does not vanish on An; it follows that gni cannot intersect An: This contradictioncompletes the proof. &

Lemma 3.2. Let f :M-Rnþ1 be a smooth immersion which is locally convex on int M;and has positive curvature on @M: Then f is locally convex on M:

Proof. Using the notion of a double of a manifold [25] it can be shown that there

exists a manifold without boundary M*M and a smooth immersion f : M-Rnþ1

such that f ¼ f on M: Since f has positive curvature on @M; there is a connected

open neighborhood U of @M in M on which f has positive curvature. Since the

second fundamental form of f on U is definite, there is a continuous choice of

normal on U with respect to which it is positive definite. Therefore f is locally con-vex on U : &

Proposition 3.3. Let M be compact, and f : M-Rnþ1 be a smooth immersion whose

curvature is nonnegative everywhere and positive on @M: Then f is one-sidedly locally

convex.

Proof. First we apply Lemma 3.1 to the interior of M: since each component X ofthe set of flat points of f is compact, then X is embedded by f onto a compact convexset.

Next let P :Mn-M be the universal cover of M; define F : Mn-Rnþ1 by F : ¼f 3P; and note that the restriction of F to the interior of Mn also satisfies thehypotheses of Lemma 3.1. Therefore, F embeds each component Y of the set of flatpoints of F onto a convex set. In particular, Y is embedded onto PðYÞ: Further,since PðYÞ lies in a component X of the set of flat points of f ; and X is path-connected, it easily follows that PðY Þ ¼ X : So Y is isometric to a compact convexset.

But then the set of nonflat points of F is connected. Indeed, a simply connectedmanifold is separated by a closed subset only if it is separated by a component of thatsubset [29, Corollary 2]. And a component Y of the flat set of F ; being isometric to a

compact convex set and lying in the interior of Mn; does not separate Mn:Since simply connected manifolds are orientable, we may choose a continuous unit

normal vector field n : Mn-Sn corresponding to the immersion F : Since the set ofnonflat points of F is connected, we may assume that n points in the positivedirection at every nonflat point. That is, the second fundamental form of F withrespect to n is positive semidefinite in a neighborhood of each point. Consequently,


the restriction of F to the interior of Mn is one-sidedly locally convex. Hence so is therestriction of f to the interior of M:

Finally, since f is smooth and is one-sidedly locally convex in the interior of M;then by Lemma 3.2, f is locally convex on M: Clearly f is one-sidedly locallyconvex. &

Now we turn to nonsmooth locally convex immersions. The following lemma isrelated to Lemma 3.1. Note, however, that Lemma 3.1 did not assume localconvexity (but rather was used to prove local convexity). Hence, the proof of thefollowing is not as subtle.

Lemma 3.4. Let f : M-Rnþ1 be a locally convex immersion, where M is complete.Let H be a local support hyperplane for f at pAint M; and X be the component of

f �1ðHÞ containing p: Suppose that XCint M: Then the restriction of f to X is an

embedding onto a closed convex subset of H:

Proof. By the definition of local convexity, each qAM has a neighborhood

Uq in M such that f embeds Uq into the boundary of a convex body KqCRnþ1:

Let ACf �1ðHÞ be the set of points in M at which H locally supports f: Bythe continuity of local support planes to a convex body [5, (1.6)], A is open and

closed in f �1ðHÞ: Therefore, H locally supports f at every point of X : Thus we maychoose Uq and Kq so that f ðUq-X Þ ¼ Kq-H; for all qAX : Therefore, every point q

of X has a neighborhood in X which is mapped homeomorphically by f onto aconvex subset of H:

But then f maps all of X homeomorphically onto a convex subset of H: Thisclaim, which completes our proof, is closely related to a classical theorem of Tietze[20,6] on local characterization of convex sets. The proof, which is quite similar tothat of Tietze’s theorem, is outlined in essentially the form we need in [29, Lemma 1],also see [19, Proposition 1]. Specifically, one shows that the set of points that can bejoined by a finitely broken H-segment to a given point pAX is open and closed.Secondly, if p and q; and q and r; respectively, lie on H-segments, then it can beshown that so do p and r: (By an H-segment, we again mean a path in M which ismapped homeomorphically by f onto a line segment in H:) Completeness of M

under distf is used to show that if an H-segment varies with one endpoint fixed and

the image of the other endpoint moving along a half-open line segment in H; then alimit H-segment exists. &

The following may be considered as the analogue of Proposition 3.3 in thenonsmooth category.

Proposition 3.5. If M is compact and f : M-Rnþ1 is a locally convex immersion that

is locally strictly convex on a neighborhood in M of @M; then f is one-sidedly locally

convex.


Proof. Let Y be the subset of M each of whose points has an open neighborhoodthat is mapped by f into a hyperplane. Then the restriction of f to M � Y has auniquely determined and continuously varying positive side.

Let Y1 be a component of Y : Then Y1 lies in a component X of f �1ðHÞ for somehyperplane H: By the strict convexity assumption at the boundary, and since X hasnonempty interior, we may conclude that XCint M: So, by Lemma 3.4, therestriction of f to X is a homeomorphism onto a compact convex set.

Then int f ðXÞ ¼ f ðint X Þ is open and convex in H: It follows from the definitionof Y1 that Y1 ¼ int X : In particular @Y1 ¼ @X : Since f ðX Þ is a compact convex set,@Y1 is connected. Since in addition, @Y1CM � Y ; it follows that the positive side off at each point of @Y1 corresponds to a fixed side of H: Thus we may extend thechoice of positive side of f continuously to Y1; and hence to all of M: Therefore, f isone-sidedly locally convex. &

Note 3.6. In Proposition 3.5, if ‘‘locally strictly convex’’ is replaced by ‘‘one-sided’’,then the proposition is no longer true; see Fig. 4. Indeed, Fig. 4 is locally convex (therequired extension across the boundary may be obtained by collaring, in each of thethree planes of the figure, the portion of the boundary in that plane) and one-sidedon a neighborhood of the boundary, but not one-sided.

Recall that in Lemma 3.4 we assumed that X-@M ¼ |; as was indeed necessaryfor the desired conclusion. We shall also require the following lemma, which allowsus to address the case in which X intersects @M:

Lemma 3.7. Let f : M-Rnþ1 be a one-sidedly locally convex immersion, where M is

complete and the restriction of f to each component of @M is an embedding. Let H be a

local support hyperplane for f at pAM; and X be the component of f �1ðHÞ containing

p: Suppose X contains every component @Ma of @M that intersects X (it is possible

that there are none). Then the restriction of f to X is an embedding onto a closed

convex subset KH of H with open subsets Da of int KH removed, where Da is a

component of H � f ð@MaÞ:

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Fig. 4.


Proof. Recall that, by definition, there is a neighborhood Up of p in M such that

fðUpÞC@Kp where Kp is a convex body lying on the positive side of f at p: By the

continuity of local support planes to a convex body, the points of f �1ðHÞ at which H

supports such a convex body Kp form an open and closed subset of f �1ðHÞ:Therefore, H locally supports the positive side of f on a fixed side of H; at everypoint of X :

Let @Ma denote a component of @M that lies in X ; so f ð@MaÞ lies in H: By thegeneralized Jordan–Brouwer separation theorem [2, pp. 353–355], H � f ð@MaÞ hasexactly two components and f ð@MaÞ is the boundary of each. We may cover @Ma byopen sets Ui :¼ Upi

chosen as above, for which @Ma-Ui is connected. Set Ki :¼ Kpi:

Choose xAint Ki; and consider the projection map pix to H through x: Specifically,

let pix 3 f be a homeomorphism of a neighborhood Uix of @Ma-Ui in Ui onto itsimage in H; where int M-Uix is connected. Define Da to be the component ofH � f ð@MaÞ that does not contain the image of int M-Uix: This choice of Da is

independent of xAint Ki-int C: Moreover, if Ui-Uja|; then one-sidedness implies

as above that int Ki-intKja|: For xAint Ki-int Kj ; the choices of Da respectively

determined by pix and pjx agree. Since @Ma is connected, it follows that the choice of

Da is independent of both x and i:

Let %M be the result of gluing the closure of Da; for each a; to M along @Ma:

Further, let %f : %M-Rnþ1 be the natural extension of f to %M: Our projection maps

and the choice of Da make clear that %M is a manifold and %f is an immersion. Let %X be

the component containing X of %f�1ðHÞ: Then %X is the result of gluing the Da to X :

By construction, %f has one-sided local supports at every point, and so %f is a locally

convex immersion of %M:

Now %XCint %M by construction. Clearly %M is complete. Therefore by Lemma 3.4,

the restriction of %f to %X is an embedding onto a closed convex subset of H: &

4. Convex caps and clippings

Here we describe the clipping procedure we mentioned in the introduction. Webegin with two lemmas that are basic tools in this paper.

Lemma 4.1 (van Heijenoort’s Theorem [18]). Let M be a manifold without boundary,

and f : M-Rnþ1 be a complete locally convex immersion. Suppose that f has a strict

local support hyperplane at some point of M: Then f is an embedding and f ðMÞ bounds

a convex body in Rnþ1:

Let KCRnþ1 be a convex body, and HCRnþ1 be a hyperplane. Suppose thatK-H has interior points in H; and let int H� be one of the open halfspaces


determined by H such that K-int H�a|: By a convex cap, cut from @K by H; wemean the closure of @K-int H�: The following lemma is due to Labourie.

Lemma 4.2 (Labourie [22]). Let M be compact and f : M-Rnþ1 be a locally convex

immersion. Let HCRnþ1 be a hyperplane which determines closed halfspaces Hþ and

H�; and set M� :¼ f �1ðint H�Þ: Suppose that f ð@MÞCint Hþ: Then f maps the

closure of each component of M� homeomorphically onto a convex cap.

Lemmas 4.1 and 4.2 are proved by hyperplane slicing arguments. The second maybe deduced from the first by a projective transformation that sends H to thehyperplane at infinity. We also need an addendum to Lemma 4.2:

Lemma 4.3. Under the hypotheses of the preceding lemma, f �1ðHþÞ is connected.

Proof. Let H*be a hyperplane obtained by moving H a small distance parallel to

itself inside Hþ: Let Hþ*

denote the halfspace containing f ð@MÞ; and Y*be the

closure of the component of f �1ðint H�*Þ containing Y : By Lemma 4.2, the closures

of the components of M� and M�*map homeomorphically to convex caps that are

pairwise nested. It follows that the closures of the components of M� lie in int M;are pairwise disjoint, and are finite in number. Moreover, each is homeomorphic to aclosed n-disk. Therefore M � M� is connected. &

Lemma 4.2 allows us to clip convex hypersurfaces by convex polyhedra, as in thefollowing proposition:

Proposition 4.4. Let f : M-Rnþ1 be a locally convex immersion of a compact

manifold M; and PCRnþ1 be a convex polyhedron. Suppose that f ð@MÞCint P: Then

there exists a locally convex immersion fP : M-Rnþ1 such that fPðMÞCP and fP ¼ f

on f �1ðPÞ: If f is one-sidedly locally convex, then so is fP: Moreover, there is a

distance-nonincreasing map from M in its f -induced metric onto M in its fP-induced

metric. We say that fP is a clipping of f by P:

Proof. We shall assume P is a closed halfspace Hþ determined by some hyperplaneH; then finitely many iterations will yield the claim. Let H

*be a hyperplane parallel

to H in Hþ; and sufficiently close to H as to satisfy f ð@MÞCHþ*: By Lemma 4.2, the

closures Yb of the components of M� :¼ f �1ðint H�Þ are mapped homeomorphically

by f onto convex caps Bb with boundary in H: Moreover, each Yb lies in the closure

of a component of M�*:¼ f �1ðint H�

*Þ which is mapped homeomorphically by f

onto a larger convex cap B*b with boundary in H

*: By the base of Bb; we mean the

convex body Db in H that is bounded by @Bb: Then there is a 1–1 correspondence

from the collection of boundary components of f �1ðHþÞ that intersect H onto the

collection of @Db; where f maps each of these boundary components of f �1ðHþÞ


homeomorphically onto the corresponding @Db: Let MH be the result of gluing the

Db to f �1ðHþÞ via these homeomorphisms. Define gH : MH-Rnþ1 by setting gH ¼ f

on f �1ðHþÞ and gH equal to the inclusion map on each Db:

Since the convex caps Bb lie in the larger convex caps B*b; it follows that gH is a

locally convex immersion of MH: Moreover, if f is one-sidedly locally convex,then the insides of the convex cap correspond to the positive side of f ; and so gH isalso one-sidedly locally convex. Since there is a homeomorphism from each convexcap Bb onto its base Db that fixes @Bb ¼ @Db; there is a homeomorphism h of M

onto MH: Thus, for P ¼ Hþ; the locally convex immersion fP ¼ gH 3 h : M-Rnþ1

satisfies all the claims of the proposition except possibly the last, which remains to beverified.

The nearest-point projection from a Euclidean space to a convex subset isdistance-nonincreasing [30, Theorem 1.2.2]. Therefore there is a distance-nonincreasing map from each convex cap Bb (in its Euclidean metric, and hence

in its f -induced metric) to its base Db: It is easily checked that this

procedure, successively applied to the finitely many convex caps Bb; yields a

distance- nonincreasing map from M in its f -induced metric onto MH in its gH-induced metric. Since the latter is isometric to M in its fP-induced metric, theproposition follows. &

Note 4.5. Lemma 4.2 and Proposition 4.4 remain valid under the weaker hypotheses

that f ð@MÞCHþ and f ð@MÞCP; respectively (as opposed to f ð@MÞCint Hþ andf ð@MÞCint P). However, these require more proof, and we do not need thesestronger statements in this paper.

5. Flat boundary components

In this section, we analyze the case in which the boundary components of a locallyconvex hypersurface lie in hyperplanes. The conclusion of the following theorem isstronger than the CHP, since in this case f is shown to be a convex embedding.Theorem 5.1 is related to a result of Rodriguez [28, Theorem 4], where f is assumedto be smooth and its restriction to each boundary component a convex embedding;but, instead of being locally convex, the surface is only required to have nonnegativecurvature; see also the paper by Kuhnel [21]. Both of these papers are based on thetheory of tight immersions and the two-piece-property of Banchoff. Our proof,however, uses van Heijenoort’s Theorem (Lemma 4.1), and assumes no smoothness.There is also a version of the following theorem for smooth surfaces withnonvanishing curvature [11].

Theorem 5.1. Let Mn be compact, and f : M-Rnþ1 be a locally convex immersion

that is locally strictly convex on a neighborhood in M of @M: Let @Ma; i ¼ 1;yk; be

the components of @M; and suppose that each f ð@MaÞ lies in a hyperplane Ha:


Furthermore, suppose that either n42; or else all nonembedded boundary components

of M lie in the same hyperplane. Then f is a convex embedding. In particular, f ðM �@MaÞ-Ha ¼ |; and M is homeomorphic to Sn with k convex open subsets removed.

The requirement that the nonembedded boundary components lie in the samehyperplane when n ¼ 2 is not superfluous, see Fig. 5(a). Also note that the abovedoes not remain valid if the surface is not locally strictly convex on a neighborhoodof the boundary, see Fig. 5(b).

Proof of Theorem 5.1. Let f : M-Rnþ1 be a locally convex extension of f :(Part 1) We show that the restriction of f to each boundary component is a locally

strictly convex immersion into the corresponding hyperplane. Note that it is onlynecessary to prove local convexity, since by hypothesis, the restriction of f to @M

contains no line segments.

For a given pA@Ma; there exists an open neighborhood U in M and a convex body

KCRnþ1 such that f embeds U into @K : Furthermore, fðU-MÞ contains no line

segments. Set K0 :¼ K-Ha; and U0 :¼ U-@Ma: Then K0 is a convex body in Ha:

Indeed, since K0 is clearly compact and convex, we need only show that K0 has

interior points in Ha: Suppose not. Then K0 lies in an affine subspace L of Ha; where

the dimension of L is less than n: Since f ðU0ÞCK0CL and f is an embedding on U0;

then by invariance of domain, the dimension of L is n � 1 and f ðU0Þ is open in L: Inparticular, a segment of any line in L passing through f ðpÞ must be contained in

f ðU0Þ; contradicting the strict convexity assumption.

(1.1) Suppose that K is not supported by Ha: Then @K-Ha ¼ @K0: Therefore

f ðU0Þ lies in the ðn � 1Þ-dimensional manifold @K0; and f jU0 is an imbedding onto

an open subset of @K0: Therefore f jU0 is a locally convex immersion into Ha; asrequired.

(1.2) Suppose K is supported by Ha: Then f ðU0ÞCK0 ¼ @K-Ha: We again claim

f ðU0ÞC@K0; from which it follows that f jU0 is a locally convex immersion into Ha:

Suppose, to the contrary, that f ðqÞ lies in the interior of K0 for some qAU0: Since aneighborhood in @K of f ðqÞ lies in Ha; then f embeds a neighborhood of q in M intoHa; in violation of the strict convexity assumption.

ARTICLE IN PRESS

(b)(a)

Fig. 5.


(Part 2) We construct an extension of M and a locally convex extension of f :(2.1) First, we show that @Ma has a neighborhood Va in M such that f ðVa � @MaÞ

lies in an open halfspace determined by Ha; say int H�a : Indeed, in both cases (1.1)

and (1.2) above, f maps a neighborhood of p in M onto an open subset of @K ; and

maps a neighborhood of p in U0 to @K0: It follows, directly in case (1.1) and from thestrict convexity assumption in case (1.2), that p has a neighborhood V in M suchthat f ðV � @MaÞ lies in an open halfspace determined by Ha: Covering @Ma byfinitely many such V yields the claim.

(2.2) Let @Mb; b ¼ 1;y; k0; denote those components of @M such that restriction

of f to @Mb; say @fb; is an embedding. Then @fb is a convex embedding by Part 1. Let

Db be the convex body in Hb bounded by @fbð@MbÞ: Since @fb:@Mb-@Db is a

homeomorphism, we may glue Db to M along @Mb via @fb; to obtain a compact,

connected manifold %M: Let %f : %M-Rnþ1 be the natural extension of f :

(2.3) We claim that %f is a locally convex immersion. Indeed, by (2.1), @Mb has a

neighborhood Vb in M such that f ðVb � @MbÞ lies in an open halfspace int H�b : It

follows that %f is locally one-to-one. Since %f is continuous, it is an immersion. Localconvexity needs to be checked only at the points where the gluing occurs. Consider

pA@Mb; and let K be a convex body such that f embeds a neighborhood of p in M

into @K : Then we are in either case (1.1) or (1.2). In the latter case, %f embeds a

neighborhood of p in %M into @K : In the former, %f embeds a neighborhood of p in %M

into the boundary of K-H�b : Thus in both cases, we have local convexity at p:

(Part 3) Now we prove that %M is a closed manifold (i.e., @M ¼ |). Suppose not.

Then, by definition of %M; @M must have some nonembedded boundary components.Consequently n ¼ 2; because if n42; then by Part 2 and van Heijenoort’s theorem(4.1), each boundary component of M is embedded. By assumption, all thenonembedded boundary components of M lie in the same hyperplane H: Since thenonembedded boundary components of M coincide with the boundary components

of %M under the inclusion map, all boundary components of %M lie in H:Let @M1 be such a component. By (2.1), @M1 has a neighborhood V1 in M such

that f ðV1 � @M1Þ lies in an open halfspace int H�: We may take V1 � @M1

connected. Let X be the closure of the component of %f�1ðint H�Þ containingV1 � @M1:

By combining van Heijenoort’s theorem with a projective transformation that

sends H to the hyperplane at infinity, we find that %f embeds int X into the boundary

in int H� of an open convex set in int H�: Let K be the closure in Rnþ1 of this convexbody. Then f ð@X Þ lies in the convex set K-H: Since f ð@M1ÞC@X by definition ofX ; and the restriction of f to @M1 is strictly locally convex by Part 1, then K-H is aconvex body in H: Thus the restriction of f to @X embeds @X onto the boundary inH of K-H: Now @M1 is a compact connected ðn � 1Þ-manifold immersed in theconnected ðn � 1Þ-manifold @X : Therefore f ð@M1Þ ¼ @X ; and @M1 is embedded,contrary to assumption.

(Part 4) Since %M is a closed manifold and %f is a locally convex immersion, then by

van Heijenoort’s theorem it follows that %f is a homeomorphism onto the boundary of


a convex body KCRnþ1: Therefore f is a homeomorphism onto the complement in@K of int Da; a ¼ 1;y; k; where Da is a convex body in Ha:

By (2.1), a given Da has a neighborhood W in %M such that %fðW � DaÞ lies in an

open halfspace int H�a : By global convexity, it follows that %fð %M � DaÞ lies in int H�

a :

Therefore f ðM � @MaÞ-Ha ¼ |: &

6. Radii of convexity: uniform estimates

In this section we start by establishing some basic tools for studying locally convexhypersurfaces, namely radius of convexity and inradius of convexity. They will be usedin this section to show that the clipping procedure developed in Section 4 satisfiescertain uniform bounds, and in the next section to formulate a convergence andfiniteness theorem.

Throughout this section, we let f : M-Rnþ1 be a one-sidedly locally convex

immersion, with locally convex extension f : M-Rnþ1: For pAM and R40; we

denote by Up;R ¼ Up;Rð f Þ; the component containing p of f�1ðint BRð f ðpÞÞ: For R

sufficiently small, f embeds Up;R into the boundary of a convex body which, by

intersection, we may take to lie in BRð f ðpÞÞ: Indeed, for R sufficiently small, there is

a convex body Kp;R; lying in BRð f ðpÞÞ and on the positive side of f ; such that f

embeds Up;R homeomorphically onto @Kp;R-int BRð f ðpÞÞ: These conditions

uniquely determine Kp;R ¼ Kp;Rð f Þ; which we call the convex piece for f centered

at p of (Euclidean) radius R: Note that we do not assume Up;R is simply connected.

The radii of convex pieces centered at p form an interval, as follows from Lemma6.1.1 by taking p ¼ q: We denote the supremum of radii of convex pieces at p byRpð f Þ; the radius of convexity of f at p. Furthermore,we set

Rð f Þ :¼ infpAM

Rpð f Þ;

the radius of convexity of f :Corresponding to each RoRð f Þ; we define the inradius of convexity rp;Rð f Þ of f at

p to be the inradius of the convex piece Kp;R: It follows by homothety that rp;Rð f Þ=R

is nonincreasing in R for RAð0;Rð f ÞÞ: (See Lemma 6.1.3 for a more generalinequality, which yields this one when p ¼ q:) The inradius of convexity rRð f Þ of f isdefined by

rRð f Þ :¼ infpAM

rp;Rð f Þ:

Note that these notions depend on the choice of extension f; which we fix in ourarguments.


Lemma 6.1. Let R2 be the radius of a convex piece centered at q: Suppose

BR1ð f ðpÞÞCBR2

ð f ðqÞÞ; and pAUq;R2: Then

(1) R1 is the radius of a convex piece centered at p;(2) Kq;R2

-BR1ð f ðpÞÞCKp;R1

;

(3) rp;R1ð f ÞX R1

R2þdð f ðpÞ;f ðqÞÞ rq;R2ð f Þ:

Proof. Let K be the convex body Kq;R2-BR1

ð f ðpÞÞ: Then fðUp;R1Þ is a component of

fðUq;R2Þ-int BR1

ð f ðpÞÞ: Let x be an interior point of K; and px : @K-@BR1ð f ðpÞÞ be

the homeomorphism given by projection from x: Then if a sequence in fðUp;R1Þ

converges to a boundary point of fðUp;R1Þ in @K ; its image sequence under px

converges to the same point. It follows that @BR1ð f ðpÞÞ is homeomorphic to the

union of fðUp;R1Þ and the complement of its px-image in @BR1

ð f ðpÞÞ: The body

bounded by this union is locally supported everywhere, hence convex, and satisfiesthe defining conditions of the convex piece Kp;R1

: Therefore 1 holds. By construction,

Kp;R1satisfies (2).

For any ball of radius d and center x contained in the convex body Kq;R2; the cone

from f ðpÞ over this ball also lies in Kq;R2: The extreme distance dn from f ðpÞ in this

cone satisfies

dn ¼ dð f ðpÞ; xÞ þ dpdð f ðpÞ; f ðqÞÞ þ dð f ðqÞ; xÞ þ dpdð f ðpÞ; f ðqÞÞ þ R2:

It follows by homothety with center f ðpÞ that a ball of radius dR1=ðdð f ðpÞ; f ðqÞÞ þR2Þ lies in Kq;R2

-BR1ð f ðpÞÞ: Therefore (2) implies (3). &

Next we introduce convex bodies K 0p;R that vary lower semicontinuously in p: The

Kp;R lack this property, as may be seen in Fig. 6.

Proposition 6.2. For RoRpð f Þ; there is a convex body K 0p;RCKp;R defined by

K 0p;R ¼ lim

e-0Kp;Rþe:

If Rð f Þ40; then for any RoRð f Þ; the convex bodies K 0p;R vary lower semicontinuously

with p: That is, for any convergent sequence K 0pi ;R

where pi-q; we have

K 0q;RC lim

i-N

K 0pi ;R

:

Proof. By Lemma 6.1.2, the convex bodies Kp;Rþe-BRð f ðpÞÞ increase monotoni-

cally as e decreases, which proves the first claim of our proposition.For the second claim, fix a sequence ei-0: Since K 0

pi ;R¼ lime-0 Kpi ;Rþe; then we

may choose a sequence di-0 such that limi-N K 0pi ;R

¼ limi-N Kpi ;Rþdi: By passing


to a subsequence of the pi; we may also assume that BRþdið f ðpiÞÞCBRþei

ð f ðqÞÞ: Then

K 0q;R ¼ lim

i-N

Kq;Rþei¼ lim

i-N

Kq;Rþei-BRþdi

ð f ðpiÞÞ

C limi-N

Kpi ;Rþdi¼ lim

i-N

K 0pi ;R

:

Here, the inclusion is by Lemma 6.1.2. The second equality is by the followingproperty of compact convex bodies: if Ci-CCB and Bi-B; then Ci-Bi-C: &

Proposition 6.3. Suppose that M is compact. Let PCRnþ1 be a convex polyhedron

such that f ð@MÞCint P: Let fP be the corresponding clipping of f ; as defined in

Proposition 4.4. Then Rð fPÞXRð f Þ=2:

Proof. Let pAM; and U ¼ Up;R=2 be the component of f �1P ðBR=2ð fPðpÞÞÞ which

contains p: Further, let Mþ :¼ f �1ðPÞ: There are two cases to consider: either (i)

U-Mþ ¼ |; or (ii) U-Mþa|:(Case i) Suppose U-Mþ ¼ |: The convex embedding of P in Rnþ1 has infinite

radius of convexity, hence has a convex piece of radius R=2 at every point by Lemma6.1.1. By assumption, fP has the same convex piece of radius R=2 centered at p:

(Case ii) Suppose U-Mþa|: Let qAU-Mþ: Then clipping by P does not reducethe radius of convexity at q: Rqð fPÞXRqð f Þ ¼ R: Thus R is the radius of a convex

piece for fP centered at q: We have BR=2ð f ðpÞÞCBRð f ðqÞÞ: Since q lies in Up;R=2; then

p lies in Uq;R (where Uq;R and Up;R=2 are both defined relative to the map fP). Now the

claim follows from Lemma 6.1.1, applied to fP: &

Proposition 6.4. Let PCRnþ1 be a convex polyhedron such that f ð@MÞCint P:

Suppose M is compact, f is one-sided, and int Ca|; where C :¼ conv f ð@MÞ: Then

there exist R40; l40 and d40; all independent of P; such that if distHðP;CÞpl; then

rRð fPÞXd:

To prove this we use the following lemma.

ARTICLE IN PRESS

f(p)f(M)

K'p, R

R

K p, R

p, R+εK

Fig. 6.


Lemma 6.5. Let M be compact and f : M-Rnþ1 be a one-sidedly locally convex

immersion whose restriction to each component of @M is an embedding. Suppose that

int Ca|; where C :¼ conv f ð@MÞ: Let f ðpÞA@C; and Up be a neighborhood of p in M

such that fðUpÞC@Kp; for some convex body KpCRnþ1 lying on the positive side of

f ðUpÞ: Then Kp-int Ca|:

Proof. First note that, for pAf �1ð@CÞ; the condition Kp-int C ¼ | is meaningful in

the sense that it holds for every choice of Up and Kp if it holds for any choice. This is

because f ðpÞAKp-C; Kp-C is convex, and the side of f on which Kp lies is

determined.Let X 0 be a component of the set

fpAf �1ð@CÞ j Kp-int C ¼ |g:

Since the condition Kp-int C ¼ | is both open and closed on the points pAf �1ð@CÞ;it follows that X 0 is also a component of f �1ð@CÞ: Since @MCf �1ð@CÞ; then X 0

contains every component of @M that it intersects.

We must show X 0 ¼ |: For any pAX 0; we denote by Hp a hyperplane that

separates Kp and C; that is, Kp and C lie in the opposite closed halfspaces determined

by Hp; say CCHþp and KpCH�

p : Since C and Kp are both convex, the existence of a

separating hyperplane follows from a well-known theorem in convexity.Suppose there exists p0AX 0; and set H ¼ Hp0 for some choice of Hp0 : Let X be the

component of f �1ðHÞ containing p0: Then (as in the proof of Lemma 3.7) thepositive side of f is locally supported on a fixed side of H at every point of X : SinceC lies on the other side of H; it follows that ðX-@MÞCX 0:

We denote by @Ma; any component of @M that intersects X : Then X 0 intersects@Ma; so X 0*@Ma: Now observe that the condition

int@C ðKp-@CÞa|

is open and closed on the points p of X 0:Moreover, since @Ma is compact, there mustexist a point qA@Ma such that f ðqÞ is contained in the interior of no line segment inf ð@MaÞ; for instance, we may let q be a point which is a maximizer for the distance of

f ð@MaÞ from any fixed point in Rnþ1: It follows that int@CðKq-@CÞa|; for,

otherwise, f ð@Ma-UqÞ would lie in an affine subspace of Hq and f ðqÞ would lie in

the interior of a line segment in f ð@MaÞ: Therefore int@C ðKp-@CÞa| for all

pA@Ma: Consequently, the choice of Hp is uniquely determined for all pA@Ma: But

then Hp is locally constant and hence constant on @Ma; so Hp ¼ H: Therefore

f ð@MaÞCH; and @MaCX : We conclude that X contains every component @Ma of@M that intersects X : Therefore by Lemma 3.7, the restriction of f to X is anembedding onto a compact convex subset KH of H with finitely many open subsetsDa of int KH removed, where Da has boundary f ð@MaÞ:

Let H� denote the halfspace determined by H that does not contain C: Then X

has a neighborhood V in M such that f ðV � X ÞCint H�: We denote by Ht; a


hyperplane parallel to H and lying in H� at distance t from H; and by H�t ; the side

of Ht that lies in H: For t sufficiently small, we may choose V to be a component of

f �1ðHþt Þ: But by Lemma 4.3, f �1ðHþ

t Þ is connected. It follows that @MCV ; andhence @MCX : In particular, f ðXÞ is a convex body in H:

Let Yt be the closure of the component of f �1ðint H�t Þ that intersects V : By

Lemma 4.2, f embeds the Yt onto nested convex caps as t-0: Therefore f embedsthe union of the Yt onto a convex cap with boundary f ð@X Þ: Thus ð

SYtÞ,X is a

compact manifold with boundary, and M ¼ ðS

YtÞ,X : But then f ð@MÞCH; in

contradiction to the assumption int Ca|: &

Proof of Proposition 6.4. Choose RAð0;Rð f Þ=4Þ: Set C :¼ conv f ð@MÞ; and MC :¼f �1ðCÞ: By Lemma 6.5, for the immersion f ; the condition K 0

p;R-int Ca| holds for

every pAMC : Further, by Proposition 6.2, the convex bodies K 0p;2R vary lower

semicontinuously in p: Together, these facts imply that for every p in some

neighborhood W in M of MC ; there is a Euclidean ball of radius d040 inK 0

p;2R-int C; where d0 is independent of p:

Now define

Mn

e :¼ f pAM j distf ðp;MCÞXeg;

where distf denotes the intrinsic distance induced on M by f : Let

lðeÞ :¼ infpAMn

e

distð f ðpÞ;CÞ;

where dist denotes the standard distance in Rnþ1: Then lðeÞ40 because Mne is

compact and f ðMne Þ-C ¼ |:

Now choose e040 so that M � Mne0CW ; and set l0 ¼ lðe0Þ: Let P be any convex

polyhedron such that CCint P; and distHðP;CÞpl0: By Proposition 6.2,Rð fPÞ42R:

Set

Mþ :¼ f �1P ðint PÞ:

Since MþCW ; then for the immersion f there is a Euclidean ball of radius d040 in

K 0p;2R-int C; and hence in Kp;2R-int C; where d0 is independent of pAMþ: But since

CCint P; it follows from the definition of the clipping fP that Kp;2R-int C is the

same for fP as it is for f : Therefore rp;2Rð fPÞXd0 for all pAMþ:

Let URðpÞ be the open distance ball about pAM with radius R in M; with respectto the metric induced by fP: As in the proof of Proposition 6.3, we consider two

cases: either (i) URðpÞ-Mþ ¼ |; or (ii) URðpÞ-Mþa|:Case (i): Suppose URðpÞ-Mþ ¼ |: Then fPðURðpÞÞCP; and consequently

rp;Rð fPÞ ¼ inradiusðBRð fPðpÞÞ-PÞ: Choose 0ol1oR; and set

d1 :¼ inffinradiusðBRðxÞ-CÞ j distðx;CÞpl1g:


Note that d140; because BRðxÞ-C depends continuously on x: Now ifdistHðP;CÞpl1; we have

rp;Rð fPÞ ¼ inradiusðBRð fPðpÞÞ-PÞ

X inradiusðBRð fPðpÞÞ-CÞXd1;

where d1 is by definition independent of P:

Case (ii): Suppose URðpÞ-Mþa|: Let qAURðpÞ-Mþ: Then rq;2Rð fPÞXd0; whererecall that d0 depends only on some fixed neighborhood W of MC in M: Inparticular, d0 is independent of P: By Lemma 6.1.3, rp;Rð fPÞXd0=3:

Thus we have proved the proposition for 0oRoRð f Þ=4; l :¼ minfl0; l1g; andd ¼ minfd0=3; d1g; which are all positive and independent of P: &

7. Convergence and finiteness theorem

Here we prove a noncollapsing convergence theorem for a sequence of locally

convex immersed hypersurfaces, fk : Mk-Rnþ1: This result occupies a place betweenthe Blaschke selection theorem and the Cheeger–Gromov compactness theorems forabstract Riemannian manifolds. Namely, it is an extrinsic analogue of the‘‘Convergence theorem of Riemannian geometry’’ ([27, p. 300]). Condition 1 belowplays the role of the bounds on injectivity radius and sectional curvature in thattheorem. Theorem 7.1 has been used recently by Guan and Spruck [16] in their workon the existence of surfaces of constant curvature with prescribed boundary.

A corollary of Theorem 7.1 is a bound on the number of homeomorphism classesof locally convex, compact immersed hypersurfaces having fixed boundary anduniform bounds on radius of convexity, inradius of convexity and volume. Bycontrast, under the special boundary conditions of Theorem 1.2, we have at mosttwo such classes, independently of such bounds. Note that finiteness theorems forLipschitz homeomorphism classes can be derived in a more abstract Gromov–Hausdorff setting; see [27, p. 299], with reference to papers of Siebenmann andSullivan [7]. However, our setting is suited to a simpler approach.

Our framework for Theorem 7.1. allows a highly intuitive (if unavoidablytechnical) proof. The ingredients include: convergence of local convex pieces, byBlaschke selection; locally defined vector fields radiating from the centers of inradiusballs in the limit convex pieces; and patching together these local vector fields toobtain a Lipschitz vector field vk that is strongly transverse to fk: Thus we use theLipschitz structure of locally convex immersions, and our uniform radius ofconvexity bounds, to construct uniform, locally embedded collars for the fk: For k

sufficiently large, this collar structure yields explicit bi-Lipschitz homeomorphismsand convergence of immersions.


For each fk; we fix a locally convex extension fk : Mk-Rnþ1: Set dk :¼ distfk;

and dk :¼ dkjMk � Mk: Recall that a l-net in a metric space M is a collection ofpoints piAM such that every point of M is at distance less than l from some pi:

Theorem 7.1. Let Mk be a sequence of compact n-manifolds whose boundaries @Mk;

which we assume nonempty, are homeomorphically identified. Let fk : Mk-Rnþ1 be a

sequence of one-sidedly locally convex immersions that are fixed on the boundary.Assume:

(1) there is a uniform lower bound R40 for the radii of convexity Rð fkÞ; such

that the corresponding inradii of convexity rRð fkÞ also have a uniform lower

bound r40;(2) for any l40; the number of elements in some l-net for the metric dk on Mk has a

uniform upper bound.

Then there exists a one-sidedly locally convex immersion f : M-Rnþ1 with locally

convex extension f : M-Rnþ1; and neighborhoods Nk of Mk in Mk; such that after

passing to a subsequence, there are homeomorphisms fk : M-Mk with bi-Lipschitz

extensions hk:M-Nk (Lipschitz constants depending only on R and r), such that

fk 3 hk-f :

We begin with an elementary lemma:

Lemma 7.2. Suppose K is a convex body in Rnþ1; and BrðyÞCKCBRðxÞ: Then:

(1) for some y0 ¼ y0ðR; rÞ40; and for any z1A@K ; K contains the truncated

spherical cone

Tz1 :¼ f z2ARnþ1 j Jz2 � z1Jpr;+z2z1ypy0g;

(2) projection from y determines a distance-nonincreasing, bi-Lipschitz home-

omorphism py : @K-@BrðyÞ; where the Lipschitz constant is determined by R

and r:

Proof. Since KCB2RðyÞ; without loss of generality we take x ¼ y:

1. Set y0 :¼ sin�1ðr=RÞ ¼ half the angle subtended by BrðyÞ from a point on@BRðyÞ: Clearly this is the required angle bound.

2. Given a segment g in BRðyÞ; we denote its length by c and its angle subtended aty by a: Since d@K is induced from Euclidean distance, it suffices to show that there isan upper bound in terms of R and r for the ratio c=a taken over all sufficiently shortsegments g whose chords do not enter BrðyÞ: For a given c; the smallest subtendedangle, say aðcÞ; occurs when g lies on the boundary of the cone over BrðyÞ from apoint on @BRðyÞ and g has an endpoint at the cone point. As in part 1, the cone angle


is 2y ¼ 2 sin�1ðr=RÞ: Since tan aðcÞ ¼ c sin y=ðR � c cos yÞ; it follows that:

limc-0

c=aðcÞ ¼ limc-0

c=tan aðcÞ ¼ R2=r: &

Proof of Theorem 7.1. Let fpki g be a finite l-net for dk on Mk ðk ¼ 0; 1;yÞ; where l

will be specified below. By condition 2, we may assume the number of points in fpki g;

say mðlÞ; is independent of k: By condition 1, for each fk there is a convex piece ofEuclidean radius R centered at every point of Mk: In particular, there is such a

convex piece Kki for fk centered at pk

i : By definition, this means that Kki is a convex

body in BRð fkðpki ÞÞ; and fk embeds an open neighborhood Uk

i CMk of pki onto

@Kki -int BRð fkðpk

i ÞÞ: In this proof, we shall refer to the neighborhood Uki as a

convex piece of Mk:

By condition 1, each Kki contains some ball Brðyk

i Þ: Since intrinsic distance is not

less than extrinsic distance, Uki contains the open dk-ball in Mk of radius R centered

at pki : Note that two different points of Mk having the same fk-image lie at

dk-distance at least R from each other.By passing to iterated subsequences, we may assume the following net convergence

properties. For each fixed i; the sequence fkðpki Þ converges to a point xiARnþ1; the

sequence yki converges to a point yiARnþ1; and the convex bodies Kk

i converge to a

convex set Ki in BRðxiÞ:Moreover, for each i and j; we may assume that the distances

dkðpki ; pk

j Þ converge. These follow from condition 2 and the assumption of fixed

boundary, which together confine the images of the fk to a bounded region. Since Ki

contains BrðyiÞ; Ki is a convex body. Further we may assume, after passing to a

subsequence, that Kki contains Br=2ðyiÞ; for all i and k:

Let Si ¼ @Ki-int BRðxiÞ; so Si is an embedded convex hypersurface in the interior

of the ball BRðxiÞ: Let dUki(resp. dSi

) denote the intrinsic distance on Uki (resp. Si).

Let BkR=2ðpk

i Þ (resp. BSi

R=2ðxiÞ) denote the corresponding closed distance balls. If

qk; rkABkR=2ðpk

i Þ; then qk and rk are joined by a dk-minimizer whose fk-image

cannot touch the boundary of BRðxiÞ: Therefore dkðqk; rkÞ ¼ dUkiðqk; rkÞ ¼

d@Kkið fkðqkÞ; fkðrkÞÞ: Suppose fkðqkÞ-z1; fkðrkÞ-z2; where qk; rkABk

R=2ðpki Þ and

z1; z2ASi: Then dSiðz1; z2Þ ¼ d@Ki

ðz1; z2Þ: Since d@Kkið fkðqkÞ; fkðrkÞÞ converges to

d@Kiðz1; z2Þ (see [5, p. 81] or [3]), we have the following distance convergence formula:

dkðqk; rkÞ-d@Kiðz1; z2Þ:

It is convenient to define a Lipschitz atlas, on an open subset of Mk containing Mk;

by using the central projections pi : Rnþ1 � Br=2ðyiÞ-@Br=2ðyiÞ from the points yi; or

what is the same, the nearest-point projections to Br=2ðyiÞ: Specifically, by Lemma

7.2.2, the map pki ¼ pi 3 fk: Bk

R=2ðpki Þ-@Br=2ðyiÞ is a distance-nonincreasing, bi-


Lipschitz homeomorphism, with Lipschitz bound depending only on R and r; from

BkR=2ðpk

i Þ; under the metric dk; onto its image in @Br=2ðyiÞ:Now we specify a net size l; depending only on r and R; as follows. For all k and i;

the image piðBSi

R=2ðxiÞÞ contains a ball of some radius 5l40 about piðxiÞ in @Br=2ðyiÞ:Thus loR=10: Let Vk

i (resp. V 0ki ) be the corresponding preimage in Bk

R=2ðpki Þ under

pki of the open ball of radius 3l (resp. 2l) about piðxiÞ in @Br=2ðyiÞ: We may

assume Bklðpk

i ÞCV 0ki and Bk

2lðpki ÞCVk

i CBkR=2ðpk

i Þ: Regarding the V ki as chart

neighborhoods, with bi-Lipschitz chart maps onto balls in @Br=2ðyiÞ; we obtain a

Lipschitz atlas on the union of the V ki carrying the metric dk: The transition

functions are bi-Lipschitz with constants depending only on R and r: Moreover, on

chart neighborhoods, dk is bi-Lipschitz equivalent to the chart distance, by Lemma

7.2.2. Therefore, since l is a Lebesgue number for the atlas fV ki g; then dkðp; qÞol

implies

dkðp; qÞoc0jfkðpÞ � fkðqÞj; ð7:1Þ

where c0 depends only on R and r:

Since pki is distance-nonincreasing, we may pull back an appropriate function on

the 3l-ball about piðxiÞ in @Br=2ðyiÞ by fk; to obtain a Lipschitz function fki :

V ki -½0; 1 ; taking value 1 on V 0k

i and 0 on the complement of a neighborhood of V 0ki

whose closure lies in V ki ; where the Lipschitz constant depends only on R and r: Then

we may define a vector field vk as follows. Let vki : V k

i -Rnþ1 be the radial vector field

vki ðpÞ :¼ jfkðpÞ � yij�1ðfkðpÞ � yiÞ: Lemma 7.2.2 states that vk

i is Lipschitz. Therefore

vk ¼ S0pipmðlÞfki vk

i is a Lipschitz vector field defined on the union of the V ki : Since

the Lipschitz constants of the fki and vk

i depend only on R and r; as does the number

of summands mðlÞ by condition 2, then so does the Lipschitz constant of vk: Thus by

(7.1), if dkðp; qÞol then

jvkðpÞ � vkðqÞjpc1dkðp; qÞpcoc1jfkðpÞ � fkðqÞj: ð7:2Þ

By the assumption of one-sided local convexity, at any point p in the union

of the Vki ; the convex hull of the rays determined by the unit vectors fvk

i ðpÞ : pAV ki g

is an outward-pointing convex cone, which is supported by any support

hyperplane for fk at p: By Lemma 7.2.1, there is a positive lower bound, y0; forthe angle between this cone and any support hyperplane at p: Therefore there is apositive lower bound, depending only on R and r; for the length of vkðpÞ for any p in

the union of the V 0ki :

Let Nk denote the union of the V 0ki ; and define Gk:Nk � ½�e; e -Rnþ1 by

Gkðp; tÞ :¼ fkðpÞ þ tvkðpÞ:

Now we prove:


Claim ðwÞ. There exist dAð0;RÞ and e40 such that for every k and every pANk; the

restriction of Gk to BkdðpÞ � ½�e; e is an embedding.

By (7.1), we may choose d0 satisfying 0od0ominðl; rÞ; independently of k and

pANk; so that BklðpÞ contains a convex piece Uk

p of Mk of Euclidean radius d0: That

is, there is a convex body KkpCBd0ðfkðpÞÞ such that fk embeds Uk

p onto

@Kkp-int Bd0ðfkðpÞÞ: Since d0ol; we have Kk

p*ðKki -Bd0ðfkðpÞÞ whenever pAV k

i ;

and hence whenever fki ðpÞa0: Therefore by Lemma 7.2.1, for any i such that

fki ðpÞa0; the convex body Kk

p contains the truncated spherical cone with vertex

fkðpÞ; radius d0; angle y0 and axis vector �vki ðpÞ: Therefore Kk

p contains the convex

hull of these truncated cones. It follows that there is a uniform d; 0odod0; suchthat Kk

p contains a truncated cone Tkp with vertex fkðpÞ; radius d; angle y0 and axis

vector �vkðpÞ: In particular, if dkðp; qÞod; then the angle between the line LkðpÞ :¼ffkðpÞ þ tvkðpÞ j tARg and the vector fkðqÞ � fkðpÞ is at least y0:

Now, writing z ¼ fkðpÞ; Claim ðwÞ may be rephrased as follows. For vectors

z1; z2; v;wARnþ1; suppose jv � wjpc0c1jz2 � z1j; and the angle y between v and z2 �z1 satisfies 0oy0pypp� y0: Then there exists e40 such that js2j4e whenever z2 þs2v2 lies on the line fz1 þ s1v1g: To verify this, we may work without loss of

generality in R2; with z1 ¼ ð0; 0Þ; v ¼ ða; 0Þ; z2 ¼ tðcos y; sin yÞ where t40; and z2 þs2w ¼ ðb; 0Þ: Then the second component of v � w is �tsiny=s2; and so

t sin y0=js2jpt sin y=js2jpc0c1t:

Thus we may take e ¼ sin y0=c0c1; and Claim ðwÞ is verified.Now we specify a new uniform net size l0 ¼ d=10; for d as in ðwÞ: Recall

doloR=10: We pass to a subsequence, also denoted by fk; with l0-nets fpki g

satisfying the net convergence properties.We may assume the Euclidean Hausdorff distance between the convex bodies Ki

and Kki is as small as we please, uniformly in k and i; and similarly for the

hypersurfaces fkðBkR=2ðpk

i ÞÞ and BSi

R=2ðxiÞ: We have seen that for pANk; the line

LkðpÞ :¼ ffkðpÞ þ tvkðpÞjtARg is the axis of a truncated spherical cone Tkp of uniform

size, where Tkp has vertex fkðpÞ and lies in Kk

p : Moreover, by construction, vkðqÞ is

arbitrarily close to v0ðpÞ if fkðqÞ is sufficiently close to f0ðpÞ: It follows that if

pAB0dðp0

i Þ; then we may assume the segment ff0ðpÞ þ tv0ðpÞj � eptpeg strikes

fkðBkR=2ðpk

i ÞÞ once only, at an angle bounded away from zero uniformly in k and i:

Therefore each pki has a neighborhood W k

i ; where Bkl0ðp

ki ÞCW k

i CBkR=2ðpk

i Þ; for

which fkðW ki Þ is the G0-image of a section of B0

2l0ðp0i Þ � ½�e; e : Moreover, this

section has Lipschitz height function over the zero section, with Lipschitz constantindependent of i and k:


Thus there is a bi-Lipschitz homeomorphism hki : W 0

i -W ki ; where W 0

i ¼ B02l0ðp

0i Þ:

Redefine the neighborhood Nk of Mk in Mk by Nk :¼S

i W ki : We now define a map

hk of N0 onto Nk by setting hkðpÞ :¼ hki ðpÞ if pAW 0

i : To see that hk is well-defined

(and hence continuous), observe that if pAW 0i -W 0

j ; then d0ðp0i ; p0

j Þo4l0: Therefore

we may assume dkðpki ; pk

j Þo5l0 and dkðhki ðpÞ; hk

j ðpÞÞo10l0oR: Therefore hki ðpÞ and

hkj ðpÞ lie on a convex piece of Mk with fkðhk

i ðpÞÞ ¼ fkðhkj ðpÞÞ; hence hk

i ðpÞ ¼ hkj ðpÞ:

The reverse argument shows that if hkðpÞ ¼ hkðqÞ then d0ðp; qÞo10l0 ¼ d; and so

p ¼ q: Therefore hk is one-to-one. Thus hk is a bi-Lipschitz homeomorphism of N0

onto Nk; with uniform Lipschitz constants. Moreover, we have constructed a map

Fk:Nk-N0 � ½�e; e for which Fk 3 hk is a section of the bundle N0 � ½�e; e : Now the

image of the limit section (which exists by construction) is our manifold M; and the

restriction of G0 to M is our immersion f: The composition of the projection from M

to N0 with hk give the desired bi-Lipschitz equivalences of M with Nk; by abuse of

notation, we denote these maps by hk as well. &

Note 7.3. In Theorem 7.1, we may substitute for condition 2, the condition that theMk have uniformly bounded volume. Indeed, if condition 2 fails, then for some l40;the number of disjoint balls of radius l=2 in Mk has no uniform bound. It easilyfollows from condition 1 and Lemma 7.2.1 that there is no uniform bound onvolume.

8. Global convexity; gluings

The following proposition is key to proving the CHP, since a limit of clippings willsatisfy its hypotheses:

Proposition 8.1. Let f : M-Rnþ1 be a locally convex immersion of a compact

manifold M; and suppose that:

(1) f is one-to-one on each component @Ma of @M;(2) f ð@MÞ lies on the boundary of a convex body C;(3) @M has a neighborhood V in M such that f ðVÞCC;(4) f is one-sided on V :

Then f is a convex embedding.

Proof. Let Ga :¼ f ð@MaÞ: @C is homeomorphic to Sn�1: So, by the generalizedJordan–Brouwer separation theorem [2, pp. 353–355], @C � Ga has exactly two

components, say D�a and Dþ

a ; and Ga is the boundary of each:

@C � Ga ¼ Dþa ,D�

a :


(Part 1) First, we show how D�a and Dþ

a may be distinguished from each other. Let

f be a locally convex extension of f to a manifold M without boundary. Cover @Ma

by finitely many open sets Ui in M such that: fjUi is an embedding on the boundary

of a convex body Ki in Rnþ1; where Ki lies on the positive side of fðUiÞ; andUi-int MCV is connected.

Case (i): Suppose Ki-int Ca| for some Ui; and hence for all Ui: (Here we are

using the fact that the condition Kp-int C ¼ | is both open and closed on the points

pA@M:) By convexity, for any xAint Ki-int C; the rays through x determine a

homeomorphism pix of @Ki onto @C; where the restriction of pix to fðUi-@MÞis the identity. Define D�

a to be the component of @C � Ga that does not contain

the connected set pixð f ðUi-int MÞÞ: Since pix varies continuously withxAint Ki-int C; this choice of D�

a is independent of xAint Ki-int C: Moreover,

the choice of D�a is independent of i: To see this, suppose pA@M-Ui-Uj: Since

fðUi-UjÞ lies on @ðKi-KjÞ; then some neighborhood in Ki of fðpÞ lies in Ki-Kj :

Since fðpÞ lies in C and Ki-C is a nonempty convex body, then every neighborhood

in Ki of fðpÞ intersects int C: Therefore int Ki-int Kj-int Ca|: For

xAint Ki-int Kj-int C; the choices of D�a respectively determined by pix and pjx

agree. Since @Ma is connected, it follows that the choices of D�a determined by any

choices of i and j agree.

Case (ii): Suppose Ki-int C ¼ | for all Ui: Since f ðUi-MÞCKi-C; thenf ðUi-MÞ lies in a hyperplane Hi separating Ki and C: Furthermore, Hi isindependent of i; say Hi ¼ H: Thus @Ma has a connected neighborhood U 0 in M

which is embedded by f into @C-H: Let D�a be the component of @C � Ga that does

not contain f ðU 0,int MÞ:Thus we have defined D�

a for every a:(Part 2) Now let %M be the closed manifold obtained by gluing the closure of D�

a ;

for each a; to M along @Ma: Further, let %f be the natural extension of f to %M: It

suffices to show that %f is locally convex at the points of @Ma: Then %f is locally convex

everywhere, and by the theorem of van Heijenoort (Lemma 4.1), %f; and consequentlyf ; is a convex embedding.

Let pA@Ma-Ui: In case (ii), %f is locally convex at p because a neighborhood of p

in %M is embedded on @C: In case (i), let %Ui be a neighborhood of p in %M; where%Ui-MCUi-M: Then %fð@Ma- %UiÞ lies on the boundary of the nonempty convex

body int Ki-int C; and %fð %UiÞ lies in C:

Our choice of D�a implies that %fð %Ui-int MÞ and %fð %Ui-D�

a Þ do not intersect, since

pixð f ðUi-int MÞÞ-D�a Þ ¼ |: Therefore %f is one–one on %Ui: Moreover, %fð %UiÞ is the

graph of a continuous radial distance function in polar coordinates about

xAint Ki-int C: Thus for a sufficiently small Euclidean ball B about %fðpÞ; %fð %UiÞseparates B into two components. Let Bþ be the closure of the component that

contains all segments joining xAint Ki-int C-int B to the points of %fð %UiÞ-B: For B

sufficiently small, BþCC: Suppose a line segment g with endpoints in Bþ leaves Bþ:Since g does not leave C or B; then g must first leave Bþ and last reenter Bþ by


leaving and entering Ki at points of f ðUi-MÞ: This is impossible since Ki is convex.

Therefore %f is locally convex at p; as required. &

9. Proof of the main theorems

We begin by showing that the strong CHP follows from the CHP:

Proposition 9.1. Suppose all locally convex immersions f : M-Rnþ1 (M compact)that fulfill conditions 1–3 of Theorem 1.2 possess the CHP. Then those that are locally

strictly convex on a neighborhood of the boundary possess the strong CHP.

Proof. We assume int Ca|; since otherwise the strong CHP holds by Theorem 5.1.

Suppose f : M-Rnþ1 satisfies conditions 1–3, and f is locally strictly convex on a

neighborhood in M of @M: By assumption, f ðMÞ-int C ¼ |: Assume, toward acontradiction, that there exists a point pAint M such that f ðpÞA@C: Then f locallysupports C at p; either on the positive side of f (case (i)), or on its negative side (case(ii)). In case (i) there is a hyperplane H that supports C and locally supports the

positive side of f at p; both in the same side, say Hþ: In case (ii), there is a hyperplaneH that supports C in one side, say H�; and locally supports the positive side of f at p

in Hþ:

In either case, let XCM be the connected component of f �1ðHÞ containing p: Bythe strict local convexity assumption, XCint M: By Lemma 3.4, f embeds X onto aclosed convex subset of H: Then X has a neighborhood U in int M; where U is a

manifold with boundary satisfying f ðUÞCHþ and f ð@UÞCint Hþ: Let d be the

distance of f ð@UÞ to H; and let H*be a hyperplane in int Hþ that is parallel to H at

distance less than d:Case (i): Denote by H�

*; the side of H

*containing H: Now we define a locally

convex immersion f*of M that agrees with f on the complement of U-f �1ðH�

*Þ:

Namely, we clip the restriction of f to U by Hþ*; as in Proposition 4.4. if H

*is

sufficiently close to H then f*ðMÞ intersects int C: But this is a contradiction, because

f*and M satisfy conditions 1–3 of Theorem 1.1, and therefore possess the CHP.

Case (ii): In this case, we denote by Hþ*; the side of H

*containing H: By

construction, the component of f �1ðint Hþ*Þ containing p lies in U ; and hence in

int M: Moreover, f ð@MÞCint Hþ*: Therefore f �1ðHþ

*Þ is not connected, in contra-

diction to Lemma 4.3.

We need the following elementary lemma; for a proof, see [30, p. 55].

Lemma 9.2. Let KCRnþ1 be a bounded convex set. Then for every e40 there exists a

convex polyhedron PCRnþ1 such that KCint P and distHðK ;PÞpe:


Proof of Theorems 1.1 and 1.2. If f satisfies the hypothesis of Theorem 1.1, then, byProposition 3.3, f is one-sidedly locally convex. Furthermore, since by assumption f

has positive curvature on @M; f is locally strictly convex on a neighborhood in M of@M: Thus f satisfies all the hypothesis of Theorem 1.2. So it remains to proveTheorem 1.2.

Suppose that f satisfies the hypothesis of Theorem 1.2. Then, by condition 2,either f is one-sided or is locally strictly convex on a neighborhood in M of @M: Butrecall that, by Proposition 3.5, strict convexity on the boundary implies one-sidedness. So we may assume that f is one-sided.

First assume int Ca|: By Lemma 9.2, for every kAN; there exists a convex

polyhedron Pk such that CCint Pk; and distHðPk;CÞp1k: Let fk :¼ fPk

be the clipping

of f by Pk; as defined in Proposition 4.4. By Propositions 6.3 and 6.4, the radii ofconvexity of fk have uniform bounds. Further, recall that, by Proposition 4.4, fk isdistance-nonincreasing. Hence, by Theorem 7.1, we obtain a limit immersion of M;say f 0: Since f 0 satisfies the hypotheses of Proposition 8.1, f 0 is a convex embedding.That is, f 0ðMÞ lies on the boundary of its convex hull: f 0ðMÞC@ conv f 0ðMÞ: Sof 0ðMÞ-int conv f 0ðMÞ ¼ |; and it follows that f 0ðMÞ-int conv f 0ð@MÞ ¼ | as well.

But note that f 0ð@MÞ ¼ f ð@MÞ: Thus we have, f 0ðMÞ-int C ¼ |: Therefore, sinceby construction f 0ðMÞ-int C ¼ f ðMÞ-int C; it follows that f ðMÞ-int C ¼ |:Further, if f is locally strictly convex on a neighborhood in M of @M; then, by

Proposition 9.1, f ðint MÞ-C ¼ |; which completes the proof of the CHP.The auxiliary result with regard to the topology of M now easily follows. To see

this, it is enough to note that f 0ðMÞC@C; because by construction f 0ðMÞCC; and,

as we showed above, f 0ðMÞ-int C ¼ |: Thus, since f 0 is an embedding andf 0ð@MÞ ¼ f ð@MÞ; M is homeomorphic to the closure of a component of @C �f ð@MÞ:

Now consider the case int C ¼ |: Then @M lies in a hyperplane H; and C is aconvex body in H: By Theorem 5.1, if f is locally strictly convex on a neighborhoodin M of @M; then f satisfies the conclusions of Theorem 1.2. It only remains to showthat if f is merely one-sidedly locally convex, then f is an embedding on @M:Moreover, M may still be regarded as being homeomorphic to the closure of acomponent of @C � f ð@MÞ with boundary f ð@MÞ; provided that we interpret @C tobe the homeomorph of a doubly covered n-disk.

Take a component @M1 of @M: Then (i) there exists a neighborhood V of @M1 inM such that f ðV � @M1Þ lies in an open half-space int H�; or (ii) there exists a point

pA@M1 and a neighborhood U of p in M such that H is a supporting hyperplane for

fðUÞ: In case (ii), let X be the component of f �1ðHÞ which contains @M1: Then X

contains every component @Ma of @M that intersects X : By Lemma 3.7, therestriction of f to X is an embedding onto a compact convex body KH of H withopen subsets Da of int KH removed. If there are no such Da; then @M1 ¼ @Ma and f

embeds M into C; so we are done. Otherwise, X has a neighborhood V in M suchthat f ðV � XÞ lies in an open halfspace int H�:

In either case, let Ht; 0otoe; be the hyperplane parallel to H in H� at distance t

from H: For t sufficiently small, let H�t be the halfspace determined by Ht that lies in


H�; and Yt be the closure of the component of f �1ðint H�t Þ that intersects V : By

Lemma 4.2, f embeds the Yt onto nested convex caps as t-0: Therefore f embedsthe union of the Yt onto a convex cap, with boundary f ð@M1Þ in case (i), and withboundary f ð@X Þ in case (ii). In case (i),

SYt is a compact manifold with boundary

@M1; and M ¼S

Yt: In case (ii), ðS

YtÞ,X is a compact manifold with boundaryS@MaCX @Ma; and M ¼ ð

SYtÞ,X : In either case, the restriction of f to @M is an

embedding. &

Note 9.3. When @M is connected, the image of @M limits the topology of M to atmost two choices by the generalized Jordan–Brouwer separation theorem [2, pp.353–355]. If @M has more than one component, then the topology of M is uniquelydetermined, because then @C � f ð@MÞ has at most one connected component that isbounded by f ð@MÞ: If @M has only one component but that component is

homeomorphic to Sn�1; then the topology of M is again uniquely determined.Indeed, if the closure of each component of @C � f ð@MÞ is a manifold withboundary, then each is homeomorphic to a ball by the generalized Schoenfliestheorem.

Acknowledgments

The authors are grateful to R. Bishop for his careful reading of drafts of thispaper, and to the referee for suggestions that improved its exposition. Further, M.G.acknowledges the hospitality of the Mathematics Department at University ofCalifornia at Santa Cruz, where part of this work was completed.

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The convex hull property and topology of hypersurfaces with nonnegative curvature