Convolution bodies and their limiting behavior

Vol. 87, No. 1 DUKE MATHEMATICAL JOURNAL (C) 1997

CONVOLUTION BODIES ANDTHEIR LIMITING BEHAVIOR

ANTONIS TSOLOMITIS

1. Introduction. Throughout this paper K and L denote convex symmetricbodies in IRn. Our notation will be the standard notation that can be found, forexample, in [7] or [9]. We start with the following definition.

Definition 1.1. For 0 < 6 < 1, the convolution body of parameter 6 of theconvex symmetric bodies K and L is the set

C(6;K,L) {x e lRn: voln(K (x + L)) > 6 voln(K L)}.

Also, if there exists a normalization exponent > 0 such that the limit

is a nondegenerate set, we call this limit the limitin# convolution body of K and L,and we denote it by C(K, L).Note that the boundaries bd(C(6; K,L)), 0 < < 1, are the level sets of the

standard convolution of the characteristic functions ;tr(x) and ;t.(x) of the setsK and L.We understand convergence as convergence (in Hausdorff sense) of the inter-

sections of our sets with any (fixed) Euclidean ball in IRn. By deoenerate set wemean a set with an empty interior. The case of an infinite cylinder is consideredto be a nondegenerate case. For every 0 < 0 < 1/2 there are examples for whichthe limiting convolution body collapses to a point for all normalization expo-nents < 0, and it converges to an infinite cylinder (or lRn), for all > 0 (seeExample 3.14).

It is immediate that the 6-convolution body C(6; K, L) as well as the limitingconvolution body C(K,L) of K and L are convex symmetric bodies (the con-vexity is a consequence of the Brunn-Minkowski inequality) and C(6; K,L)=C(6; L, K), C(K, L) C(L, K). For K L the convolution bodies and their limit-ing behavior was studied by Schmuckenschliger in [8] where he proved that if

Received December 1995.The author’s work was partially supported by a National Science Foundation grant.

181

182 ANTONIS TSOLOMITIS

voln(K) 1 and 0 < 6 < 1, then

limC(6; K, K) HOK,

6--1- 1 --6

where IIK denotes the projection body of K, and we write I/K for the polar ofHK. Let us recall here that the polar I/K of the projection body of K is the con-vex body in IRn whose corresponding norm is given by

Ilullrioc lul voln-l(proj[u]+/- (K)),

for all u e IR", where l" is the standard Euclidean norm in IR" and [u] is the1-dimensional subspace of IR" generated by u. For more information on the pro-jection body, see, for example, the recent survey [1] or the book [9]. We note herethat the corresponding relation for the volumes, i.e.,

limvoln(C(5; K, K)) voln(iiOK)

51- (1 -6)"

was earlier proven by Kiener in [4], and later Schmuckenschliiger in [8] provedmore:

(1 6)IIK_

C(6; K, K)_ log () IIK.

This theorem can give information about the projection body HK through prop-erties of the di-convolution bodies C(6; K,K). For example, the 6-convolutionbodies of affinely equivalent convex bodies are easily seen to be themselves affinelyequivalent. Hence the same is true for the polar of the projection bodies of affinelyequivalent bodies. Consequently, we have the following.

FACT 1.2. If the convex symmetric bodies K and L are affinely equivalent, thenthe same is true for their projection bodies IlK and IlL.

Fact 1.2 is not obvious from the definition of HK (see, for example, the "non-trivial" proof in 1 ]).

Meyer, Reisner, and Schmuckenschlfiger continued the study of convolutionbodies in [5]. In their paper, they study properties of the convolution procedurein the case that the two bodies are the same.To illustrate the different possibilities that may occur in the case that K and

L are different, let us mention a few examples of limiting convolution bodies,which we compute in Section 2.For 1 < p < c and 0 < 6 < 1, set s (n+p- 1)/p (s 1 ifp az). Then the

limiting convolution body of Be and Be?, is homothetie to Be: with normal-ization exponent 1Is (see Example 2.1).

Another example is that of eonvolving the body Be7 with (1/n)Be. The limit-

CONVOLUTION BODIES AND THEIR LIMITING BEHAVIOR 183

ing convolution body exists with normalization exponent cz l/n, and it inducesin IRn a norm "almost isometric" to the el-norm (see Example 2.3).The case L

_K so that L is "well fitted" in K (for the exact definition of "well

fitted," see Definition 2.4) is presented in the Theorem 3.2. This theorem statesthat if the common boundary bd(K) c bd(L) of K and L is "rich enough" (i.e., itsprojection on every hyperplane has a positive (n- 1)-dimensional volume), thenthe limiting convolution body of K and L exists with normalization exponent

1. The theorem also provides us with a formula for the norm of C(K,L),which resembles the formula that defines the norm of the polar of a projectionbody. Let us also note that the case K L studied in [8] is included in Theorem3.2 as a special case.

In contrast to all previous examples and results, one of our main theorems(Theorem 3.9) states that if convex symmetric bodies K and L are "generic intheir relative position" (see Definition 3.7), then the limiting convolution body isan ellipsoid (possibly degenerated; see Remarks 3.10 and 3.11). The normal-ization exponent is 1/2. The theorem provides us with a formula for thenorm (possibly a seminorm) corresponding to C(K, L) as well.The limiting convolution body should be interpreted as the indicatrix of a rel-

ative change of volume in different directions in IRn. From this perspective, theexamples that we computed demonstrate in a very exact, quantitative way non-stability of volume behavior in high dimensions. However, Theorem 3.9 showsthat for bodies that are "generic in their relative position" the relative volumechange in different directions is surprisingly regular.Note that one of our main results, Theorem 3.9, was announced in [10].We want to thank Professor V. D. Milman for his encouragement and guid-

ance in this research.

2. Convolution body and examples. The following example describes thelimiting convolution body C(K, L) in the case that K is the unit ball of/? andL is the unit ball of for 1

EXAMPLE 2.1. Let l < p < c, O < cS < l, and set s (n + p-1)/p (s l ifp ). Then we have

C(Be2o Beg) limC(3;Beo,Bq) An,pBq,

where An,p are constants dependin# on n and p. They are uniformly bounded andseparated from zero for n e ]hi and 1 < p < , and their exact value is

An,p1

2,

1/s

forp#


Remark 2.2. It is interesting to note that convolving the unit ball of withitself gives (up to homothety) the unit ball of ’. On the other hand, although

,tn_l) < n 1/(n- , 1 +enn_l is almost isometric with (Banach-Mazur-d(e n 1)

(log n)/n ), when we convolve their unit balls the result is (up to homothety) theunit ball of/?.

Proof. For p the result follows from the result in [8], since it is wellknown that the projection body of Be is (1/2)Be. Assume 1 < p < . Letx= (Xl,X2,...,Xn) be a point on the boundary of C(6;Be,Bq). If 6 is closeenough to 1, x will satisfy

voln((x + Be,)\Beoo) voln-l((1 tp)I/PBeg-)dtj=l 1-1xjl

(1 3) voln (Beg),

which implies

voln-l(Bq-) (1 tP) ("-t)/p dt (1 t) voln(Bq).i-1 1-1xjl

For every e [1- Ixjl, 1] by the intermediate value theorem, there exists t e(1 Ixl, 1) so that 1 tp ptp-1 (1 t) and the above becomes

voln-l(Be,-,)j=l 1-1xl

p(n-1)/pn-1)(p-1)/P(1 t) (n-1)/p dt (1- 6)voln(Be;).

Using the information on t, Bernoulli’s inequality, and finally evaluating theintegrals, we get

1_ (n- 1)(P- 1)P

[xjl (n/p-1)/p

Since all xj converge to zero as 6 1, we get our conclusion. [-1

EXAMPLE 2.3. The convolution body C(3; Be, (1/n)Be) satisfies

( 1 ) C(,;Be,(1/n)Be)C Be7,Be llim (1 -6)/cnL


where

and

211n(n!)l/ncn= 1.

Proof. Let X=(X1,X2,...,Xn) be a point on the boundary of C(6;BeT,(1/n)Beo). For 6 dose enough to 1, we can assume that x+ (1/n)Be\Be7 consistsof 2"-1 n-dimensional simplices. Each of them has one (n- 1)-dimensional faceon one of the (n 1)-dimensional faces of BeT. The simplex with "base" on the faceof Be7 orthogonal to the vector (el/X/-h, e2/x/-h,...,e,/x/) has height equal to

(1/V/-)lelXl +/32X2 -+-"’" d- enXn], where ej + 1, Vj 1,2,... ,n. Each 1-dimen-sional side of each such simplex (the ones parallel to the axis) has length equal to

Consequently,

1 1ejxj (1 -6/,vol ( r\(x + g

from which we get

2n+ln!

The result now follows. [-1

We continue by introducing first some terminology.

Definition 2.4. Let K and L be two convex symmetric bodies in IRn. We saythat the body L is "well fitted" in the body K if the following two conditionshold:

(1) LK,(2) Vx lRn\(0}, we have x + L K.

Observe that the number of touching points of the boundaries of K and L in


the above definition can even be two. This is the case, for example, for the bodiesBq and the ellipsoid (centered at the origin) with one semiaxis of length equalto 1 and all other semiaxes of length equal to 2.We have the following.

PROPOSITION 2.5. Let K be a convex symmetric polytope such that Bq is wellfitted in K. Then the limiting convolution body C(K, Bq) ofK and the Euclideanball is a section of Bt+)/2, for some rn IN.

Proof. Let _Zl, ___z,..., +_Zm (for some me N) be the touching points ofbd(K) and n-. A similar computation as the computation in Example 2.1 givesthat if x is a point on the boundary of the convolution body C(i; K, Bq), then itsatisfies

lim(jI[(XZj)[ (n+l)/2)

2/(n+l)

(1 6)/(+)(n + 1) 2/cn+1) (voln(Bq) )2/(.+1)2

Thus

C(5; K, Bqlim cnL,a+l(1 _a)2/(n+l)

where

L= x e [(x, zj)[ (n+)/2 1j=l

THEOREM 2.6. Let K be any convex symmetric body in which Bq is well fitted.Then for every e > O, there exists a convex symmetric body P in which Bq is wellfitted, so that d(K,P) < 1 + e and C(P, Bq) is almost isometric (up to nun) to amultiple of an affine position of Beo

Proof. It is enough to show that the body P can be chosen so that it has only2n touching points _+z, _z2,..., _+zn with the Euclidean ball, with z,z2,...,

zn being linearly independent, and in a neighborhood of each zj, j 1, 2,..., n theboundary of P is flat. Then the result will follow from an analogous computationlike in Proposition 2.5.

Let k > 1 be the maximal number of linearly independent vectors on the com-mon boundary of K and Bq. Let z1, z2,..., Zk be such a set. Let 0 > 0 and eo > 0be such that (1 + eo)2 < 1 +e and (1 + (O/n))" < 1 + eo. Choose Zk+l Nn-lc(span{z2, za,...,Zk}) +/- so that Zk+l is linearly independent of Zl and so that if


K1 K {x e lit"" I(Zk+l, X)l 1} we have

0d(K, KI) < I +-.

This choice is possible since as Zk+ gets arbitrarily close to z, K1 gets arbitrarilyclose to K (in the Hausdorff distance). We continue this procedure (choosing thenext Zk+2 -Nn-c (span{z2,z3,...,Zk}) -t- so that Zk+2, Zk+, and z are linearlyindependent, and if K2 KI {x IRn" I(Zk+2,x>l < 1}, then d(K1,K2) <(1 + (O/n)), until we end up with a body K Kn-k that its common boundarywith Bg has n linearly independent vectors Zl,Z2,... ,Zn, and

d(K,g) < 1 + < 1 + eo.

Now let s > 0 and set

n

/s (1 + s)/ I’ {x e IR"" I(zg, x)l < 1}.j=l

It is clear from the definition of Ks that in the Hausdorff distance

lim/ =/.s--+0

Choose so small enough so that d(/,/s0) < 1 + e0, and put P =/s0.

3. The main results. We start by introducing some additional notation.For y on the boundary bd(K) of K, we denote by Nr(y) (or N(y) if it is clear

to which body we refer) the normal vector of bd(K) at y, if it exists and isunique. By convexity the normal vector exists for a subset of bd(K) of full mea-sure (see [9]).For a function from a set A to IR, we denote by 0+ the function

where ;rE denotes the characteristic function of the set E.Before we state our next theorem, let us note that the norm of the polar of

projection body of the body K is also given by the following:

1 Jb I<N(Y), u)l d2(y),

where 2 is the Lebesgue measure restricted on bd(K). The last expression showsthat the projection body is always a zonoid. A modification of the previousexpression will appear in the next theorem. Let K, L be two convex symmetric


bodies in IRn so that L K. Then the quantity

(NK(y), u)+ d2(y),d(K)cbd(L)

for USn-1 defines, by homogeneity, a norm (the triangle inequality by((N(y),u) + (N(y),u2))+ < (N(y),u)+ + (N(y),u2)+; the index K can bedropped from the vectors NK(y) since the above integral does not change if Nr isreplaced by NL), provided that bd(K) c bd(L) is "rich" enough.

FACT 3.1. Let K be any convex symmetric body in lRn, and let A be any subsetof bd(K), satisfying

Then for x lRn, the quantity

Ixl vol._ (proj[x] {y A" (N(y), x) > 0)).

where Ixl denotes the Euclidean norm of x, defines a norm.

The above norms can be obtained through the mixed convolution bodies.

THEOREM 3.2. Let K, L be two convex symmetric bodies, so that L is well fittedin K. Assume that bd(K) c bd(L) has countably many components and

2((y e bd(K) bd(L)" (N(y), u) > 0}) > 0 Vu

where 2 denotes the Lebesoue measure of lRn restricted on bd(K) c bd(L). Then thelimitino convolution body C(K, L) satisfies

C(K, L) limC(6; K, L)

6--}1- 1 -6

and its norm on n-1 is 9iven by

1(2) Ilullc< , ./- voln(L)

voln_l(proj[u]x {y bd(K) c bd(L)" (N(y), u) > 0}),

for all u e -1.Remark 3.3. If K L, then the above result gives that the limiting convolu-

tion body of K is the polar of its projection body providing another proof forthis result of [8].


Proofof Theorem 3.2. Let us assume that

bd(K) c bd(L) U( +- Uj),

where the + U/s are the components of bd(K) c bd(L).Let u -, and for 0 > O, let k0 ko(O) be such that

A (prj[ul (j=ko+l(+--- UJ))) < O"

For y > 0, set

Uj,r (x e b(K)" dist(x, Uj) <

where dist is, say, the geodesic distance on bd(K). Clearly,

(3) lim 2(Uj,r\Uj) 0y--,O

for all j N.For 0 < t < 1, we construct the following truncation of L: consider the cylin-

der F {lRu} + L (Minkowski addition) and set Lt (tF)c L. Lt is a convexsymmetric body,

(4) proj[u], F proj[ulL

and

(5) lim Lt L

(say, in the Hausdorff distance). By convexity and compactness, it follows as wellthat

(6) lim voln_ (proj[u] (b(L)\Lt)) 0

and

(7) min {length((y + lRu) c (b(L,) \b(L))} > 0.yeb(Lt)\b(L)

Thus, if is close enough to 1 (i.e., 6(t, y) < di < 1 for some 6(t, y) > 0), we


have

(1 6)voln(Lt) voln

and our construction shows

I < (1- )llullc< ; C,La voln(Lt) < I + II,

where

f| (N(y), u)+ d2(y)bd(K)bd(L,)

and

(N(y), u)+ d2(y).

The last inequalities can be rewritten as

I Ilullc(6; ,,.a/o-6) voln(Lt) < I + II.

Letting now 6 1- the lim inf and the lim sup of the norm are boundedbetween I and I + II.

Let y go to zero, and then let 0 go to zero. Then the integral II will convergeto zero (since the domain of integration converges measurewise to zero and theintegrand is a bounded function). This proves that the limit of the latter normexists and equals the integral I. Thus we have arrived at

(8)1 I (N(y), up d2(y).[[UlIc(K’Lt) voln(Lt) d(K)mbd(L,)

We want now to let 1-. The right-hand side of (8) converges to

voln(L) d(K)cbd(L)(N(y), u)+ d2(y),

since all operations involved are continuous, and this equals

voln(L) voln-l(proj[u]+/- {y bd(K) c bd(L)" (N(y), u) > 0}).

It remains to show the following claim.


CLAIM. We have lim_-Ilullc(r,L,) --Ilullcc,.Proofof the claim. Allow volume (1 6) voln(L) to go out of K by shifting L

in the direction of u. Say that the shift is achieved by the vector tau. Assume alsothat volume (1- 6)voln(Lt) gets out of K by a shift of Lt by the vector flau(fl fl(t)). Define for convenience the following quantities:

voln(L\Lt)fba(r)rb,(t,) (N(y), u)+ d2(y)

A=I+voln-1 (proj [u] x (ba(L)\L,))

fba(r)cba(L,) (N(y), u)+ d2(y)

If L is shifted by the vector (fl + (1 6)F)u, the volume that gets out of K isat least

(1 6) voln(Lt) + (1 6)I" Jbd(K)cbd(Lt) (N(y), u)+ d2(y),

and this equals (1 6) vol,(L). Hence we conclude that

(9) a < fla + (1 6)F,

provided that 6 is close enough to one.Similarly, if Lt is shifted by (aA (1 6)F)u, the volume that gets out of K is

at least

(1 6)voln(L) 06 voln-l(proj[u]l (bd(L)\Lt))

+ (o(A 1) (1 6)F) / (N(y),u)+ d2(y),Jbd(K)cbd(Lt)

which equals (1 6)vol,(Lt). Consequently,

(10) fla < aaA- (1 -6)F,

provided that 8 is close enough to 1.Combining (9) and (10), we get

(11) fla + (1 6)1" < oa < fla + (1 di)F.


We observe that

(1 -)llullc<;,.)-

and similarly for fl. Letting di 1- in the relation (11) we get that the quantitieslim infa--Ilullcla;:,)/l-a) and lim supn-l-Ilullca;r,)/ll-a) are bounded betweenthe quantities

A+F]C K,L

and

Since this is true for all 0 < < 1, letting 1-, and noting that F goes to zeroand A to 1, we get that the limit of as 1- exists and is equal tothe limt__,l-IlUllc(r,L,), proving the claim and the existence of the limiting con-volution body of K and L as well. [--]

Remark 3.4. If bd(K) c bd(L) is strictly convex, then the condition (1) is auto-matically guaranteed if we just assume that the Lebesgue measure of bd(K c L)is positive. Of course, this is also the case when K or L is strictly convex.

Remark 3.5. In the case that K and L are not symmetric, Theorem 3.2 isvalid although the relation (2) does not define a norm but the Minkowski func-tional of C(K, L).COROLLARY 3.6. Let K Bn be the Euclidean ball in lRn of radius 1. Let A be

a closed subset of $n-1, symmetric with respect to the origin and such that2(A) > O. Then, for every invertible linear operator T: IR -- IP, we have

vOln_(prOj[rul+/-(T({x A" (NTI(Tx), Tu) > 0})))voln_l(proj[u]+/-{x

_A" <x,u>/> 0))

where u IRn and l" denotes the Euclidean norm.

Proof. Let L be the convex hull of A. Then, since for every invertibleT: IRn IRn we have

T(C(K,L)) C(TK, TL),

we conclude that


Now we get the result by using the special form of the norm of C(K,L) andC(TK, TL) described in Theorem 3.2.

We continue with the theorem for the ease that the two bodies K and L havean (n- 2)-dimensional manifold as the intersection of their boundaries. This, ina sense, may be considered to be the "general ease."

Definition 3.7. The convex symmetric bodies K and L are said to be "genericin their relative position" if they satisfy the following conditions:

(1) voln((x + K) c L) < voln(K c L) for all x e lRn\{0}.(2) The set M bd(K)c bd(L) is an (n- 2)-dimensional manifold and the

manifolds bd(K) and bd(L) are C at every point of M. Moreover, if 2 is the(n- 2)-dimensional volume measure restricted on M, then 0 < 2(M) < c.

(3) M contains no "touching points," i.e., there is no x e M for which Nr(x)

Remark 3.8. The condition (1) in Definition 3.7 is always true if "<" isreplaced by "<". This is a consequence of the Brunn-Minkowski inequality.Here, though, we require the strict inequality because we are interested in thecase that C(6; K, L) "shrinks" down to zero as 6 tends to 1; the strict inequalityguarantees exactly this.The condition (2) in the above definition implies that the outer unit normal

vectors Nr(x) and NL(X) of K and L, respectively, exist (and they are unique)for all x M, and their restrictions on M are continuous functions of x.We state now our next theorem: if Vl and v2 are two unit vectors in IR", we

write Iv1 v2l for the quantity (1- (v,v2)2) 1/.TrmoRra 3.9 (Ellipsoid limit convolution theorem). Let K and L be two con-

vex symmetric bodies that are generic in their relative position. Then the limitin9convolution body ofK and L is an ellipsoid (possibly "de#enerated"; see Remarks3.10 and 3.11). In particular,

C(6;K,L)C(K,L)= lim--)I- (1 6) /2

and

( IM )1/2(u,(*) Ilullc< , ) 2 voln(/c L) INr(x) x NL(x)[

for all u IRn.Before we proceed with the proof of Theorem 3.9 some remarks are due.

The conditions required by Theorem 3.9 are not really restrictive, as they are"usually" satisfied. For an example, consider K pBq and L Bq for 1 <p < q < , 1 < p < n(/p)-(/q), and p k(1/)-(/), k 1, 2,..., n.


Bodies whose boundaries intersect on an (n- 2)-dimensional manifold, butnot necessarily C1, can be included in some cases in the ellipsoid limit con-volution theorem through special approximation arguments. In particular, theresult of the theorem is true for polytopes (see Theorem 3.13).

Remark 3.10. It may happen that [Ic(r,L) is a seminorm, which means thatthe ellipsoid "degenerates" to an infinite cylinder. There are examples for whichthe limiting convolution body degenerates to IRn for all exponents > 1/2 andcollapses to a point for all < 1/2 (see Example 3.15).

Remark 3.11. Typically--for "generic" bodies in some sense--(,) defines anorm and not a seminorm, which means that the limiting convolution body is anondegenerated ellipsoid. However, in the case that (,) defines the zero seminorm,the behavior of the convolution bodies is not so regular: there are examples forwhich the limiting convolution body, after suitably normalizing the convolutionbody of parameter 6, can be very different from an ellipsoid, for example, a cube(see Example 3.16). In the case that (,) defines a nonzero seminorm (but not anorm), then it is not possible to receive a nondegenerated limiting convolutionbody without using different normalizations in different directions.

Remark 3.12. If the convolution f of the characteristic functions ;r and ;L ofK and L, respectively, is of class C2 at the point zero, then the indicatrix ofDupin exists for the graph of the function f at the point zero, it is an ellipsoid(see, for example, [11]), and this ellipsoid is the limiting convolution body of Kand L computed in the Theorem 3.9. However, the level of smoothness of f atzero is not known a priori. From the final stages of the proof of the Theorem 3.9,one can see that f is C at zero, although the result itself implies (through a the-orem of Aleksandrov; see [9]) that f is C2 at zero. In the case K L, the func-tion gr, g behaves as a cone at zero, and so it is not even differentiable.However, in this case the limiting convolution body can be an ellipsoid.

Before we continue with the proof of Theorem 3.9, let us introduce someadditional notation. For an arc AB we write AB’. A triangle defined by thepoints A, B, F is denoted by ABF; the angle at B is denoted by A[F, at F withAFB and so on. We write IABI for the length of the vector AB.

Let us also recall that a tubular neighborhood V of a k-dimensional manifoldM in IRn, for which a normal (n- k)-dimensional subspace Fx exists at everypoint x M, is an open neighborhood of M such that for all x, y M with x y,(Fx V) and (Fy V) are disjoint. The existence of a tubular neighborhood isguaranteed if M is a C compact manifold (see, for example, [3]).

Proof of Theorem 3.9.set

Let u IRn and let Uj, j IN, be the components of the

{x bd(K) c L" (u, Nr(x)) > 0}.

Shifting K by the vector u0 u/l[U[[c(6;t(, bd(C(6; K, L)), K "captures" some


volume from L in the directions Nr(x) for x Uj, j ]N. Simultaneously, somevolume from L that was inside K is lost, and this is in a neighborhood of the -Uj’sin the direction Nr(x) for x 6 UJ, J 6 IN. Since, by symmetry

-[(K c L)\((K x) c L)] (K c L)\((K + x) c3 L),

the loss of volume from L that occurs when we shift K by u0 is the same as theloss of volume when we shift K by -u0 that occurs in a neighborhood of U in thedirections -Nr(x), x Ug. Thus, in order to compute the loss of volume

(1 )voln(K c L) voln(K L) voln((U0 / K) L),

we have only to look at a neighborhood of the U’s in the directions N(x) and-N(x) for all x U for all j IN.

Let 0 be a positive number and ULo be a 0-neighborhood of Uj on bd(K), i.e.,

Uj,o {x bd(K)" geodesic-distance (x, Uj) < 0}.

We may consider that we have only a finite number of U’s by considering only thefirst m of them, with m e IN, such that 2((.Jj_m+ Uj) is less than a given number,say e, and use an approximation argument--letting e go to zero. Then 0 can bechosen small enough so that the U,o’s are all disjoint. For 5 close enough to 1, wemay write

(1 5) voln(K c L) voln(K c L) voln((Uo + K) c L)

--/VOln(j0tlU((UJ’-tu)tZ)l-vln (O<t<lU ((Uj,o+tu)cL)II"’Y/vln(j0<t<lU ((Uj,o-tu)tZ)l-vln (O<t<lU

E(VOln(TI,j) voln(T2,j)),J

where

and

U [(Uj,o tuo) c L] \[((Uj,o + tuo)c L) uo]O<t<l

U [((Uj,o + tuo)cL)- uo]\[(Uj,o tuo)cL].O<t<l


Fix j IN and label with A a point x in the (n- 2)-dimensional manifoldMI,j--bd(U,ocL). Let B,F,A be the intersection points of the 2-dimensionalplane spanned by the vectors Nr(x) and N(x) with the (n- 2)-dimensionalmanifolds M2,j bd([(U,o + uo) L] uo), M3,j bd((Uj,o L) u0), and M4,jbd((Uj,o- uo)oL), respectively, so that the plane figure defined by the pointsA,B,F,A and the boundaries of K,K-u0,L,L-u0 on the Sx=span(Nt:(x), N(x)} is inside T, or T2,j. It is easy to see that being in T, or T2,depends on whether (Nr(x), Uo)(NL(x), Uo) > 0 or (NK(x), uo)(N(x), uo) < O,respectively. To compute the volumes voln(Tld) and voln(T2d), we will integrateon M the volumes of the plane figures ABFA as described above, for all x e M (allj’s). (Here we make use of the existence of a tubular neighborhood ofM in lRn.) Weconcentrate now our attention on the 2-dimensional section x + Sx. Let E be thepoint of intersection of the line BF with the tangent line of bd(L) at A; let Z bethe point of intersection of the line FA with the tangent line of bd(K) at A; let H bethe point of intersection of the arc AB-" with the tangent line of bd(K uo) at F;and let (R) be the point of intersection of the arc AA’-" with the tangent line ofbd(L- uo) at F. These intersection points exist for di close enough to 1, by thecompactness of M and the condition (3) of the Definition 3.7. Let us write ABFfor the region enclosed by bd(L) (x + Sx), bd(K uo) (x + S) and AF, andby AAF the region enclosed by bd(L- uo) (x + Sx), bd(K) (x + S) and AF.Clearly,

AFH" ABF m__ AFE

AF(R) _m AFA_AFZ

Thus,

vol2(AFH) + vol2(AF(R)) < vol2(ABFA) < vol2(AFE) + vol(AFZ).

Now,

vol2(AFE) -IA-II<AF, Nc(A)>I,

1vol(AFZ) }hl I<h, Nc(A)>I,

vol2(AFH) IAI I<Ar’, NAn(A)>I,

vol2(ArO) IAI I<Ar’, NAo(A))[,


where for a segment AB we write, say, Nan(B) for the unit normal vector of AB atB with a direction such that has a positive inner product with x. Note that asdi 1- the normal vectors converge to either Nr(x) or NL(x) (Nan(A) convergesto Nt(x), and NAo(A) converges to Ni(x!).

Using now the law of sines to estimate IaEI, I1, I1, and I1, we get

1 (I<A,N_.o(F)><A-, Nan(a)>l

l< A--,i_ao(_NAo(A) ><---0-iA--’NL-uo (r) >l)< vo12(ABFA)

1 (I<A,Nnr(F)><A,NL(A)>Il<A--, Nr’a(F)5<A--, NK(A)>l)

Observe that F M- u0 and tends to x as di tends to 1. To finish the proof, weneed the following claim.

CLAIM. The vector Ilullc(;,,>hF converges (say, in the Euclidean norm) to

Px(-U), where Px is the orthogonal projection of lRn to S.Now let q: M -- {__+ 1} be defined by (x) 1 if ABFA

_TI,j, j 1, 2,...,

and (p(x) =-1 if ABFA T2,j, j- 1, 2, From the above claim, it followsthat vol2(ABFA) < o(1/liull2c(a;,)). Hence, using Weyl’s theorem for comput-ing the volume of a tube (see [2]), we have

C(,5;K,L)/(I_,5)I/2VOln(K C L)-1({1})

Ilull 2 volE(ABFA)dxC(;r,L)

2 -1((_1}C(6;tC,L) vo12 (ABI"A) dx

C(tS;K,L) 0 Ilullc;,L)

where no ambiguity arises if we recall that ABFA depends on x. We now use theestimates on vol2(ABFA) and take the lim sup6_,1 and lim inf6l in the above


equation. Using the monotone convergence theorem and Fatou’s lemma, we getour result, since (Px(u),Nr(x))= (u,Nc(x)) and (Px(u),NL(x))- (u, NL(x)).

Proofofthe claim. Let P be the projection point of x + u0 on x + S,, and let Ube the terminal point of the vector x + u0. Then AP x + Px(-uo), ]AF AP]

Thus we must prove that Ilullcl;,,)[P’l converges to zero as 6-- 1. Wehave that IP--fi[ IFI cos(PPU), and using the law of sines in AFU, we get that

IFI sin(I’JU) 1

sin(A ’V) Ilullc< ;K,L)

Consequently,

sin(F U) cos(V ’U).IlulIc(6;K, .) sin(APU)

Observing that by condition (3) of Definition 3.7 A’U tends (as di 1) to 7r/2 andPFU converges to 7r/2, we finish the proof of the claim.

We continue now with the case of polytopes. For x on the boundary of a con-vex body K, we write N(K, x) for the normal cone of K at x (see [9]).

THEOREM 3.13. Let P, Q be two convex symmetric polytopes satisfying:(1) voln(P C (X + Q)) < voln(P C Q)for all x IRn\{0};(2) P and Q have no "touchin9 points," i.e., if x M bd(P) c bd(Q), then nei-

ther N(P, x)_N(Q, x) nor N(Q, x)

_N(P, x).

Then the limitin# convolution body ofP and Q, exists, and it is an ellipsoid.

For a convex symmetric body K, we will write extr K for its r-skeleton andfor a set A_IRn, x*lRn, and e>0, we will write WA(X*;e) for the set{x A: I(x*, x)l > 1 }.

Sketch of the proof of Theorem 3.13. Let Z (M c extn_2P) w(Mcext,_2Q). Clearly, M\Z is a C (n-2)-dimensional manifold. LetF1, F2,..., Fv be all the (n 2)-dimensional faces of P, and G1, G2,..., GR all the(n 2)-dimensional faces of Q for some N, R ]N. For each Fi and Gj (1 < < N,1 < j < R), let x and y be vectors in bd(P) and bd(Q), respectively, so thateach x belongs to the interior of the normal cone of Fi, each y] belongs to theinterior of the normal cone of Gj, and for all 1 < < N, 1 < j < R, the vectors xand y7 are not collinear. Consequently,

(1) eXtn-2P ----- Ufq__l Wp(X;;/) and extn-2Q -- U=I WQ(y;e),(2) 2n_1(7=1Wbd(P)(X;; ()) CF, and #n-l(O=l Wbd(Q)(Y;;e)) C,,

where 2n- and #n- denote the Lebesgue measure of lR" restricted on bd(P) andbd(Q), respectively. We now construct convex symmetric bodies Px, P2,..., PN, Pand Q, Q2,..., Qv, Q by successively truncating P and Q (Pi Pi(e) and Qj(e))


in the following way: Set first P0 P, Q0 Q, and assume that the bodies Pj-1 andQj-1 have been defined. For each vector x, we examine whether eventually (fore -, 0) the set We(x; ) ext,_Q is empty or not. (It cannot be frequently non-empty.) If the above set is eventually empty, then we set Pj P-I\Wp(xf; e) andQj Qj_l If not, then let {Y)*,I, Y)*,2,... *Yj,mj} be a subset of {y y*2, Y,} ofminimal cardinality (mj IN) so that

Wp(x;e) c extn_2Q __. U Wba()(Y)*,i; e),i=l

for all e > 0. Then we define P Pj-l\Wp(x;;8) and

Qj Qj-1 WQ(y)*,i;i=l

where 0j,i(e)> 0 are chosen so that the sets {x e Pj-I: I<x ,x>l- 1- e} andE;{Y Qj-I: ]<Yj,i, Y>] 1 e} intersect on an (n- 2)-dimensional manifold for all

’s and *’s.) We continue> 0. (Here we use the pairwise noncollinearity of the xinductively until all x, x,... ,xv are used. Finally, we construct P, and Q. SetP Pv. Let {Yq+I,1,..., YN+l,m+a} be the subset of {y]’,..., y[} that consists ofthe vectors that have not been used in the above inductive procedure. (It may beempty.) Set

mN+l

Q Qv U WQ(yI+I,mN+, )"

We now can use the same arguments used in the proof of Theorem 3.9 toprove the following inequalities. We write M for bd(P)c bd(Q) and C(6) forC(6;P,Q):

12 -’((1))\(M\M)Ilull(a)vol2(ABrA) dx " -((-1))\(M\M)

Ilull(a)vol(ABrA) dx

c0)1(1_,),12 voln-2(P r Q)

- ({1})\(M \M)

2 -’((-1})\(M\M)Ilull vOlE(ABFA)dx

200 ANTONIS TSOLOMITIS-- _1 ({1})c(M\M2) Ilull(a)vol2()j TI,j( (X+Sx)ldx-F -({-1}) c (M\Mze) Ilull(a)vol2(L)j T2,j (x+Sx))dx,

where Sx is the 2-dimensional subspace of IRn orthogonal to M2 at x. Clearly, bythe same arguments in the proof of Theorem 3.9 and the construction of M2,, thelast two integrals are less than e for some constant > 0 independent of e and 6.Passing to the limits lim sup6__. and lim inf6l and then lim,__.0, the aboveinequalities give the result.

We observe here that the result in Theorem 3.9 is much more "unstable" thanthat of Theorem 3.2 in the following sense.

THEOIM-EXAMPLE 3.14. There exists a universal constant c > 0 such that forevery 1 < p < 2, for every n > c, and for every 0 < e < 1/2, we can find convexsymmetric bodies K, L, L and points _+ el, + e2,..., _+ en on the boundary of K, sothat

(1) bd(K) cbd(L) (bd(K) cbd(L)) w {_+e, _+e2,..., +_en};(2) L

_L_

(1 + e)L;(3) C(K, L) is an ellipsoid;(4) C(K, L) is (up to homothety) Bq.Proof. Let l<r< and consider the sets L0=rBe and K0--Bet.

Assume that bd(L0) is equipped with the normalized Haar measure #. LetA r$n-\Ko. It follows (see [7]) that there exist constants c,c2 so that#(A) < ce-c2n. Thus, by a standard concentration-of-measure argument as de-scribed in [7] (i.e., using the fact that the complement of A has an exponen-tially big measure), we get that there exists a universal constant c, so that forn > c, there exists an orthonormal basis e, e2,..., en of lRn, satisfying rej Afor all j 1, 2,..., n. Let min{ 1, dist(rej, conv(A)), j 1, 2,..., n} > 0, andfor 0 < < define the sets

L(t) rBey {x_rBe" I<x, ej>l > r(1-

Let L L() and

K Bet xsBeo’l<x, ej>l > r 1--


It is clear that

bd(K) c bd(L) bd(rBe) c bd(Be),

so that C(K, L) is an ellipsoid. (Actually, it can be shown by direct computationthat it is a Euclidean ball.)

Let

L=conv L()wr 1---where q (n- 1)/(p- 1) and here the ball Be is understood as to be expressedwith respect to the basis e,e,...,en. Then, according to Theorem 3.9 andExample 2.1, for close enough to 1 and u s Nn-, we have that

lib (u’Ntc(x))(u’N(X))dx+o(,)21_C(6;K,L) (K)cb(e) INK(x) NL(X)I

(n+q-1)/q - (1 6)voln(K c L),

where Cp,n is a constant depending only on p and n and the similarity constantstend to 1 as 6 tends to 1. Using now the value of q and rearranging, we get

C(,;KL)

21lull 2C(6;K,L)

+ Cp,ny I<u, ej>l t’ - voln(r c

Since p < 2, if we let di 1-, the fraction of norms in the first side of theabove equation converges to zero; thus we get that

c(r,L)Cp,n

voln(g c L) Z. I<u, ej)l p

and

C(K, L) limC(6; K, L)

di---l- (1 t) lip

We complete the proof by observing that

1 e/2L=L= L=(I+e)L1 e

if e > 0 is chosen to satisfy 0 < e < 1/2. [-1


We now present two examples describing the situations discussed in Remarks3.11 and 3.12. Our examples are given in IRE but revolution of the bodies to bedescribed with respect to the el-axis and eE-axis in IRn will produce similarexamples in higher dimensions.

EXAMPLE 3.15. For any 0 < o < 1/2, we construct bodies K and L in ]R2 sothat

C(6;K,L) -,{0}

and

C(6; K, L) ]R2 if o > Oto,-a)

as 6 --+ 1-.

The convergence to IRE is understood as convergence of (C(3; K, L)/(1 6)) crBq to rBq, in the Hausdorff distance, for every r > 0.

Construction. For all 0 < a0 < 1/2, consider the function

dxli log for 0 < x < 1

f(t)= x0, for x < 0.

Consider now K to be the body bounded by the curves:

’y=l,

y= -1,

y l-f(x- 1),

y =-1 +f(x- 1),

y l-f(-x- 1),

y=-l+f(-x-1),

for 1 <x<2

for 1 <x<2

for-2 <x<-I

for-2 <x<-l.

Let R denote the rotation of IR2 by n/2 (say, with det(R) 1), and let L R(K).The proof that this example works is omitted.

EXAMVLE 3.16. There are bodies K and L in IR2 so that the equation (,) inTheorem 3.9 defines the zero seminorm for which a different normalization 9ives a

limitin9 convolution body equal to the 2-dimensional cube.


Construction.use the function

We use the same construction of the last example, but now we

le-1f(x)=

/x,

O,

forx> 0

for x < 0.

We omit the proof.

REFERENCES

[1] J. BOURGAIN AND J. LINDENSTRAOSS, "Projection bodies" in Geometric Aspects of FunctionalAnalysis (1986/87), Lecture Notes in Math. 1317, Springer-Verlag, Berlin, 1988, 250--270.

[2] A. GRAY, Tubes, Addison-Wesley, Advanced Book Program, Redwood City, Calif., 1990.[3] M.W. HIRSCH, Differential Topology, Grad. Texts in Math. 33, Springer-Verlag, New York,

1976.[4] K. KIENER, Extremalitiit yon Ellipsoiden und die Faltungsungleichung yon Sobolev, Arch. Math.

(Basel) 46 (1986), 162-168.[5] M. MEYER, S. REISNER, AND M. SCHMOCKENSCHLT,ER, The volume ofthe intersection ofa convex

body with its translates, Mathematika 40 (1993) 278-289.[6] V. MILMAN AND A. PAJOR, "Isotropic position and inertia ellipsoids and zonoids of the unit ball

of a normed n-dimensional space" in Geometric Aspects of Functional Analysis (1987/88),Lecture Notes in Math. 1376, Springer-Vedag, Berlin, 1989, 64-104.

[7] V. MILMAN AND G. SCHECHTMAN, Asymptotic Theory of Finite-Dimensional Normed Spaces,Lecture Notes in Math. 1200, Springer-Vedag, Berlin, 1986.

[8] M. SCHMUCKENSCHLGER, The distribution function of the convolution square of a convex sym-metric body in IRn, Israel J. Math. 78 (1992) 309-334.

[9] R. SCHNEIDER, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl. 44,Cambridge University Press, Cambridge, 1993.

[10] A. TSOLOMITIS, On the convolution body of two convex bodies, C. R. Acad. Sci. S6r. Math. 322(1996), 63-67.

DEPARTMENT OF MATHEMATICS, Tim Omo STATE UNIVERSITY, 231 WEST 18TH AVENUE, COLUMaUS,OI-no 43210, USA; [email protected]

Documents

Convolution bodies and their limiting behavior