STABILITY RESULTS INVOLVING SURFACE AREAMEASURES OF CONVEX BODIES

DANIEL HUG AND ROLF SCHNEIDER

Abstract

We strengthen some known stability results from the Brunn-Minkowski the-ory and obtain new results of similar types. These results concern pairs ofconvex bodies for which either surface area measures, or counterparts of suchmeasures in the Brunn-Minkowski-Firey theory, or geometrically significanttransforms of such measures, are close to each other.MSC 2000: 52A20, 52A40.

1 Introduction

In recent decades, several of the classical uniqueness theorems for convex bodieshave been turned into quantitative versions, in the form of stability results. Thestarting point for the present investigation are uniqueness theorems of Minkowskiand Aleksandrov, respectively. Minkowski’s theorem, in its later general form, saysthat a d-dimensional convex body is uniquely determined, up to a translation, byits (d−1)st surface area measure. A theorem of Aleksandrov, independently provedby Fenchel and Jessen, states the extension of this result to lower order surface areameasures. Aleksandrov’s projection theorem asserts that a d-dimensional convexbody with a given centre of symmetry is uniquely determined by the volumes (orintrinsic volumes of a given positive order) of its (d − 1)-dimensional orthogonalprojections. Also this theorem involves surface area measures, since volumes of(d − 1)-dimensional orthogonal projections and area measures are related by thecosine transform, see (6).

In the following, we improve some known stability results corresponding tothese uniqueness theorems, and we obtain new stability versions of some similaruniqueness assertions.

To formulate a stability version of Minkowski’s uniqueness theorem, we denoteby Kd(r, R) the set of convex bodies in Euclidean space Rd which contain some ballof radius r > 0 and are contained in some ball of radius R > r. Let K, L ∈ Kd(r, R)be convex bodies whose surface area measures Sd−1(K, ·) and Sd−1(L, ·) satisfy

|Sd−1(K, ·)− Sd−1(L, ·)| ≤ ε. (1)

A typical stability version of Minkowski’s theorem requires to find a number α > 0,which depends only on d, and a number c > 0 depending only on d, r, R such thatfor ε ≥ 0, inequality (1) implies

δ(K, L + x) ≤ cεα (2)

1

with a suitable x ∈ Rd; here δ denotes the Hausdorff metric. The best result ofthis type up to now is due to Diskant [4] (Theorem 7.2.2 in [23]); it gives a stabilityorder α = 1/d. Under the stronger assumption that K, L are of class C2

+ and that

|Sd−1(K, ·)− Sd−1(L, ·)| ≤ εσ, (3)

where σ denotes the spherical Lebesgue measure, Diskant [6] (with details in [7])obtained (2) with the better exponent α = 1/(d−1). Our first result, to be provedin Section 2, will achieve (2) with α = 1/(d− 1) under the weaker assumption (1),for a large class of convex bodies including polytopes and bodies of class C2

+. Wealso show that in this result the exponent 1/(d− 1) is optimal.

Assumption (1) is essentially equivalent to an assumption on the total variationnorm of the difference of the surface area measures of K and L: (1) implies

‖Sd−1(K, ·)− Sd−1(L, ·)‖TV ≤ 2ε, (4)

and (4) implies (1) with ε replaced by 2ε.Following a suggestion of Wolfgang Weil, we replace this assumption on the

total variation distance of measures by a more natural one on the Prohorov distancedP . The inequality (1) implies

dP (Sd−1(K, ·), Sd−1(L, ·)) ≤ ε, (5)

but not conversely. We show in Section 3 that the weaker assumption (5) is stillsufficient to obtain the stability estimate (2) for all convex bodies K, L ∈ Kd(r, R)with α = 1/d.

For the Aleksandrov-Fenchel-Jessen theorem on lower order surface area mea-sures (Corollary 7.2.5 in [23]), a stability version was obtained by Schneider [22](Theorem 7.2.6 in [23]). Lutwak’s work on the Brunn-Minkowski-Firey theory,where Minkowski sums of convex bodies are replaced by Firey’s p-sums, containsalso a generalization of the Aleksandrov-Fenchel-Jessen theorem (Corollary (2.3)in [19]). In Section 4 we will give a stability result for this extended theorem. Ascorollaries of the proof, we obtain stability versions of two inequalities of Lutwak[19].

For a convex body K ⊂ Rd and a unit vector u ∈ Sd−1, we denote by Ku theimage of K under orthogonal projection onto u⊥, the hyperplane through 0 or-thogonal to u. We write Vd−1(·) for the volume in (d−1)-dimensional hyperplanes.Then

Vd−1(Ku) =12

∫Sd−1

|〈u, v〉|Sd−1(K, dv), (6)

where 〈·, ·〉 is the scalar product of Rd. Thus, the projection function u 7→Vd−1(Ku) of K is, up to a constant factor, the cosine transform of the surfacearea measure Sd−1(K, ·). A special case of Aleksandrov’s projection theorem (e.g.,Theorem 3.3.6 in [10]) says that two d-dimensional centrally symmetric convexbodies with the same projection function differ only by a translation. Stability

2

versions of this uniqueness theorem are due to Campi [3] (for d = 3) and toBourgain & Lindenstrauss [2]. In Section 5 we use the method of Bourgain andLindenstrauss to obtain further stability results of a similar nature. One of theseresults concerns the sine transform

u 7→∫

Sd−1

√1− 〈u, v〉2 Sd−1(K, dv)

of the surface area measure, which also has geometric significance. Then we obtainsome stability results for convex bodies which are not necessarily centrally sym-metric. They refer to various integral transforms, appearing in work of Anikonov& Stepanov [1], Goodey & Weil [12], Schneider [24].

2 Stability for Minkowski’s theorem

We work in d-dimensional real vector space Rd (d ≥ 3), equipped with the standardEuclidean structure. The set of convex bodies (non-empty compact convex sets)in Rd is denoted by Kd. For notions from the theory of convex bodies which arenot explained here, we refer to [23]. Apart from replacing En by Rd, we use theterminology of that book.

Let K ∈ Kd be a convex body, and let Sd−1(K, ·) be its surface area mea-sure of order d − 1. It is a finite Borel measure on the unit sphere Sd−1. ByLebesgue’s decomposition theorem, it can be decomposed, with respect to the(d − 1)-dimensional Hausdorff measure Hd−1, into an absolutely continuous partSa

d−1(K, ·) and a singular part Ssd−1(K, ·). The latter can be decomposed further,

by definingSc

d−1(K, ·) :=∑

u∈Sd−1

Sd−1(K, {u})δu, (7)

where δu denotes the Dirac measure (unit point mass) at u, and

Snd−1(K, ·) := Ss

d−1(K, ·)− Scd−1(K, ·).

Clearly, in (7) at most countably many summands are non-zero. Moreover,Sd−1(K, {u}) = Hd−1(F (K, u)) for u ∈ Sd−1, where F (K, u) is the support set ofK with outer normal vector u. Thus, we have the decomposition

Sd−1(K, ·) = Sad−1(K, ·) + Sc

d−1(K, ·) + Snd−1(K, ·) (8)

of the surface area measure Sd−1(K, ·) into an absolutely continuous measureSa

d−1(K, ·), a component Scd−1(K, ·) which is an at most countable sum of point

masses, and a singular component Snd−1(K, ·) without point masses.

2.1 Theorem. Let 0 < r < R. There exists a number c, which depends only ond, r, R, with the following property. If K, L ∈ Kd(r, R) are convex bodies satisfying

3

Snd−1(K, ·) = 0, Sn

d−1(L, ·) = 0 and the assumption

|Sd−1(K, ·)− Sd−1(L, ·)| ≤ ε (9)

for some ε ≥ 0, thenδ(K, L + x) ≤ cε

1d−1

for a suitable vector x ∈ Rd.

The condition Snd−1(K, ·) = 0 is fulfilled, for example, if K is a polytope, or if

the surface area measure of K is absolutely continuous. The latter is true, inparticular, if the support function of K is of class C2.

The exponent 1/(d − 1) in Theorem 2.1 is optimal, at least for the class ofpolytopes. This can be seen by choosing for K a unit cube and for L the polytopewhich is obtained from K by cutting off a vertex of K in such a way that thesection plane meets K in a regular (d − 1)-simplex of edge length ε1/(d−1). Onthe other hand, under the stronger assumption that L is a ball and that (3) holds,Theorem 3.4 in [17] achieves (2) with α = 1.

The subsequent proof of Theorem 2.1 is a refinement of the approach of Diskant[6], [7] and makes also use of Diskant [5]. The proof does not require the fullcondition Sn

d−1(K, ·) = 0 = Snd−1(L, ·), but only its consequence

Snd−1((1− t)K + tL, ·) ≥ max{Sn

d−1(K, ·), Snd−1(L, ·)} for 0 ≤ t ≤ 1; (10)

hence we will work under this assumption.As usual, we write

V1(K, L) := V (K[d− 1], L)

for K, L ∈ Kd, where V denotes the mixed volume, thus

V1(K, L) =1d

∫Sd−1

h(L, u)Sd−1(K, du).

Here h(L, ·) is the support function of L. We also write V (·) for the volumefunctional in Rd.

We assume that K, L ∈ Kd(r, R) are convex bodies satisfying (9) and (10). Fort ∈ [0, 1] we set

Ht := (1− t)K + tL,

then Ht ∈ Kd(r, R). In the following, c1, c2, . . . denote positive constants whichdepend only on d, r, R.

The proof is divided into four steps.

Step I. First we show that

|V (Ht)− V (K)| ≤ c1ε, |V (Ht)− V (L)| ≤ c1ε (11)

4

for t ∈ [0, 1].

By Lemma 7.2.3 in [23], the estimates

|V (L)− V1(K, L)| ≤ c2ε, (12)

0 ≤ V1(K, L)− V (K)d−1

d V (L)1d ≤ c3ε (13)

and the corresponding ones with K and L interchanged follow from the assump-tions on K and L. From V1(K, L)d ≥ V (K)d−1V (L) and (12) we deduce that

V (L)d

(1 +

c2ε

V (L)

)d

= (V (L) + c2ε)d ≥ V (K)d−1V (L),

henceV (K) ≤ V (L) (1 + c4ε)

dd−1 ≤ V (L) + c5ε.

By symmetry, we infer that

|V (K)− V (L)| ≤ c5ε.

Since K, L ∈ Kd(r, R), this implies∣∣∣V (K)1/d − V (L)1/d∣∣∣ ≤ c6ε. (14)

The function φ defined by

φ(t) := V (Ht)1/d − (1− t)V (K)1/d − tV (L)1/d, t ∈ [0, 1],

is concave and satisfies φ(0) = φ(1) = 0, hence

φ′(0) ≥ φ(t) ≥ 0 for t ∈ [0, 1]. (15)

Using (13) and K ∈ Kd(r, R) we get

φ′(0) =V1(K, L)− V (K)

d−1d V (L)

1d

V (K)d−1

d

≤ c7ε. (16)

Hence, by (14), (15) and (16) we obtain∣∣∣V (Ht)1/d − V (K)1/d∣∣∣ ≤ ∣∣∣V (L)1/d − V (K)1/d

∣∣∣+ c7ε ≤ c8ε.

Since Ht,K ∈ Kd(r, R), this implies the first inequality of (11), and the secondfollows by symmetry.

Step II. Next we show an analogue of (11) for surface areas, namely∣∣V1(Ht, Bd)− V1(K, Bd)

∣∣ ≤ c9ε,∣∣V1(Ht, B

d)− V1(L,Bd)∣∣ ≤ c9ε (17)

5

for t ∈ [0, 1]; here Bd is the unit ball.

If the support function h(M, ·) of the convex body M ∈ Kd is second order dif-ferentiable at u ∈ Sd−1 (which holds for Hd−1 almost all u ∈ Sd−1), then theeigenvalues of the second order differential d2h(M, ·)|u⊥ at u are the principalradii of curvature of ∂M at the point with outer normal vector u. Their productis denoted by Dd−1h(M,u), thus

Dd−1h(M,u) = det(d2h(M,u)|u⊥).

For all u ∈ Sd−1 with the property that h(K, ·) and h(L, ·) are second orderdifferentiable at u, and thus for Hd−1 almost all u ∈ Sd−1, we set

m(u) := min{Dd−1h(K, u), Dd−1h(L, u)}.

Minkowski’s determinant inequality states that

Dd−1h(Ht, u)1

d−1 ≥ (1− t)Dd−1h(K, u)1

d−1 + tDd−1h(L, u)1

d−1 ,

henceDd−1h(Ht, u) ≥ m(u) (18)

for all t ∈ [0, 1]. Let ω1 denote the measurable set of all u ∈ Sd−1 such that h(K, ·)and h(L, ·) are second order differentiable at u and Dd−1h(K, u) > m(u). Thenm(u) = Dd−1h(L, u) for u ∈ ω1 and

0 ≤∫

Sd−1(Dd−1h(K, u)−m(u))Hd−1(du)

=∫

ω1

(Dd−1h(K, u)−Dd−1h(L, u))Hd−1(du)

= Sd−1(K, ω1)− Sd−1(L, ω1).

Here we have used the fact that if M ∈ Kd and ω is the set of all u ∈ Sd−1 suchthat h(M, ·) is second order differentiable at u, then the restriction of the measureSd−1(M, ·) to ω is absolutely continuous with respect to Hd−1. This can be verifiedon the basis of [15], [16] (and with the terminology used there): Choose any normalboundary point x ∈ τ(M,ω) of M , that is, x = τM (u) for a uniquely determinedu ∈ ω. Then by an inspection of the proofs of Lemma 3.4 and Lemma 3.1 in [15],we obtain ki(x, u) > 0 for i ∈ {1, . . . , d− 1}; moreover, by Lemma 3.1 in [15], wefind ki(x) = ki(x, u) for i ∈ {1, . . . , d − 1}. This shows that Hd−1(M,x) > 0 forHd−1 almost all x ∈ τ(M,ω). The assertion is then implied by Theorem 3.7 in[16].

Now it follows from (9) that

0 ≤∫

Sd−1(Dd−1h(K, u)−m(u))Hd−1(du) ≤ ε. (19)

6

For M ∈ Kd and u ∈ Sd−1, we set

f(M,u) := Vd−1(F (M,u))

andm(u) := min{f(K, u), f(L, u)}.

The additivity of support sets ([23], Theorem 1.7.5(c)), together with the Brunn-Minkowski theorem, implies that

f(Ht, u)1

d−1 ≥ (1− t)f(K, u)1

d−1 + tf(L, u)1

d−1

and thereforef(Ht, u) ≥ m(u) (20)

for u ∈ Sd−1 and all t ∈ [0, 1].Let ω2 denote the set of all u ∈ Sd−1 such that f(K, u) > m(u). Hence, ω2 is

at most countable, and for u ∈ ω2 we have m(u) = f(L, u) and

f(K, u)−m(u) = Sd−1(K, {u})− Sd−1(L, {u}),

thus

0 ≤∑

u∈Sd−1

(f(K, u)−m(u)) =∑u∈ω2

(f(K, u)−m(u))

= Sd−1(K, ω2)− Sd−1(L, ω2).

Therefore, (9) implies

0 ≤∑

u∈Sd−1

(f(K, u)−m(u)) ≤ ε. (21)

For convex bodies M,H ∈ Kd, we now make use of the decomposition

V1(M,H) =1d

∫Sd−1

h(H,u)[Sa

d−1(M,du) + Scd−1(M,du) + Sn

d−1(M,du)]

= V a1 (M,H) + V c

1 (M,H) + V n1 (M,H)

with

V a1 (M,H) :=

1d

∫h(H,u)Dd−1h(M,u)Hd−1(du),

V c1 (M,H) :=

1d

∑h(H,u)f(M,u),

V n1 (M,H) :=

1d

∫h(H,u)Sn

d−1(M,du).

7

Here, as below, we write∫

instead of∫

Sd−1 if the integration is extended over Sd−1.Similarly,

∑means

∑u∈Sd−1 , where the summation effectively extends only over

countably many summands.We estimate the expression

I := V (Ht)− V n1 (K, Ht)−

1d

∫h(Ht, u)m(u)Hd−1(du)− 1

d

∑h(Ht, u)m(u)

from both sides. Without loss of generality, we assume that rBd ⊂ K, L. Thenthe support function of Ht satisfies r ≤ h(Ht, ·) ≤ 2R. First we insert in I theexpression

−V n1 (K, Ht) = −V1(K, Ht) + V a

1 (K, Ht) + V c1 (K, Ht)

and use Ht = (1− t)K + tL together with the estimates (11), (19), (21) to obtain

I = V (Ht)−1d

∫h(Ht, u)Sd−1(K, du)

+1d

∫h(Ht, u)(Dd−1h(K, u)−m(u))Hd−1(du)

+1d

∑h(Ht, u)(f(K, u)−m(u))Hd−1(du)

≤ (1− t)(V (Ht)− V (K)) + t(V (Ht)− V1(K, L))

+2R

d

[∫(Dd−1h(K, u)−m(u))Hd−1(du) +

∑(f(K, u)−m(u))

]≤ t(V (Ht)− V1(K, L)) + c10ε. (22)

Next, we insert in I the decomposition

V (Ht) = V a1 (Ht,Ht) + V c

1 (Ht,Ht) + V n1 (Ht,Ht)

and use the fact that Snd−1(Ht, ·) − Sn

d−1(K, ·) is, by (10), a positive measure.Together with (18) and (20), this gives

I =1d

∫h(Ht, u)

(Sn

d−1(Ht, du)− Snd−1(K, du)

)+

1d

∫h(Ht, u) (Dd−1h(Ht, u)−m(u))Hd−1(du)

+1d

∑h(Ht, u)(f(Ht, u)−m(u))

≥ r

d

{Sn

d−1

(Ht, S

d−1)− Sn

d−1(K, Sd−1)

8

+∫

(Dd−1h(Ht, u)−m(u))Hd−1(du) +∑

(f(Ht, u)−m(u))}

≥ r

d

{Sn

d−1(Ht, Sd−1)− Sn

d−1(K, Sd−1)

+∫

(Dd−1h(Ht, u)−Dd−1h(K, u))Hd−1(du)

+∑

(f(Ht, u)− f(K, u))}

= r(V1(Ht, Bd)− V1(K, Bd)). (23)

Combining (22) and (23), we find

V1(Ht, Bd)− V1(K, Bd) ≤ t

r(V (Ht)− V1(K, L)) + c11ε.

By (11) and (12),

|V (Ht)− V1(K, L)| ≤ |V (Ht)− V (L)|+ |V (L)− V1(K, L)| ≤ c12ε

and thusV1(Ht, B

d)− V1(K, Bd) ≤ c13ε, t ∈ [0, 1]. (24)

On the other hand, from (18), (20), (10), (19), (21) we deduce

V1(Ht, Bd) = V a

1 (Ht, Bd) + V c

1 (Ht, Bd) + V n

1 (Ht, Bd)

≥ 1d

∫m(u)Hd−1(du) +

1d

∑m(u) +

1dSn

d−1(Ht, Sd−1)

=1d

∫Dd−1h(K, u)Hd−1(du) +

1d

∑f(K, u) +

1dSn

d−1(K, Sd−1)

−1d

∫(Dd−1h(K, u)−m(u))Hd−1(du)

−1d

∑(f(K, u)−m(u)) +

1d

(Sn

d−1

(Ht, S

d−1)− Sn

d−1(K, Sd−1))

≥ V1(K, Bd)− 2dε. (25)

Now (24) and (25) yield the first estimate of (17), and the second follows byinterchanging K and L.

The next step provides corresponding estimates for projection volumes.

Step III. If v ∈ Sd−1 and t ∈ [0, 1], then

|Vd−1 (Hvt )− Vd−1(Kv)| ≤ c14ε, |Vd−1 (Hv

t )− Vd−1(Lv)| ≤ c14ε. (26)

9

For the proof, we define

J := Vd−1 (Hvt )− 1

2

∫|〈u, v〉|Sn

d−1(K, du)

−12

∫|〈u, v〉|m(u)Hd−1(du)− 1

2

∑|〈u, v〉|m(u).

Since

J =12

∫|〈u, v〉|(Dd−1h(Ht, u)−m(u))Hd−1(du)

+12

∑|〈u, v〉|(f(Ht, u)−m(u))

+12

∫|〈u, v〉|

(Sn

d−1(Ht, du)− Snd−1(K, du)

)≥ 0,

we can deduce that

Vd−1 (Hvt )− Vd−1(Kv) ≥ −1

2

∫|〈u, v〉|(Dd−1h(K, u)−m(u))Hd−1(du)

−12

∑|〈u, v〉|(f(K, u)−m(u))

≥ −ε. (27)

On the other hand, by (17), (19), (21)

Vd−1 (Hvt )− Vd−1(Kv) ≤ J

=12

∫|〈u, v〉|

(Sn

d−1(Ht, du)− Snd−1(K, du)

)+

12

∫|〈u, v〉| (Dd−1h(Ht, u)−m(u))Hd−1(du)

+12

∑|〈u, v〉|(f(Ht, u)−m(u))

≤ Snd−1(Ht, S

d−1)− Snd−1(K, Sd−1)

+∫

(Dd−1h(Ht, u)−m(u))Hd−1(du) +∑

(f(Ht, u)−m(u))

= d(V1(Ht, Bd)− V1(K, Bd))

10

+∫

(Dd−1h(K, u)−m(u))Hd−1(du) +∑

(f(K, u)−m(u))

≤ c15ε. (28)

The estimates (27) and (28) yield the first estimate in (26), and the second estimatefollows by symmetry.

Step IV. The rest of the proof now follows from the work of Diskant [5], [7]. Forgiven v ∈ Sd−1, the function defined by

φv(K, L, t) := Vd−1 (Hvt )

1d−1 − (1− t)Vd−1(Kv)

1d−1 − tVd−1(Lv)

1d−1

for t ∈ [0, 1] can be estimated, in view of (26), by

φv(K, L, t) = (1− t)[Vd−1 (Hv

t )1

d−1 − Vd−1(Kv)1

d−1

]+t[Vd−1 (Hv

t )1

d−1 − Vd−1(Lv)d−1]

≤ c16ε.

If λ > 0 is such that Vd−1(λKv) = Vd−1(Lv), one obtains φv(λK,L, t) ≤ c17ε.Now the main theorem of [5] shows that there exist ε0 > 0 and c18, dependingonly on d, r, R, such that

δ(λKv, Lv + x(v)) ≤ c18ε1

d−1

for some x(v) ∈ v⊥, if ε ≤ ε0, and therefore

δ(Kv, Lv + x(v)) ≤ c19ε1

d−1 .

For ε > ε0, the same inequality holds if the constant c19 is adjusted. Thus theassertion of Theorem 2.1 is implied by Theorem 4.3.4 in [10].

3 Stability and Prohorov metric

For a set A ⊂ Sd−1 and for ε > 0, let

Aε := {y ∈ Sd−1 : ‖x− y‖ < ε for some x ∈ A},

where ‖ · ‖ is the Euclidean norm. For finite Borel measures µ, ν on Sd−1, let

dP (µ, ν) := inf{ε > 0 : µ(A) ≤ ν(Aε) + ε and ν(A) ≤ µ(Aε) + ε

for all Borel sets A ⊂ Sd−1}.

11

This defines the Prohorov metric dP , which metrizes the weak topology (e.g., see[8], Section 11.3, in the case of probability measures).

The following theorem strengthens Diskant’s stability theorem (Theorem 7.2.2in [23]), replacing the assumption (1) by the weaker assumption (5).

3.1 Theorem. Let 0 < r < R. There exists a number c, depending only on d, r, R,such that, for K, L ∈ Kd(r, R),

δ(K, L + x) ≤ cdP (Sd−1(K, ·), Sd−1(L, ·))1/d

for some x ∈ Rd.

Proof. In the following, the positive constants c1, c2, . . . depend only on d, r, R.Let K, L ∈ Kd(r, R). We set µ := Sd−1(K, ·), ν := Sd−1(L, ·), µ1 := µ(Sd−1),ν1 := ν(Sd−1), ε := dP (µ, ν). Then

|µ1 − ν1| ≤ ε, µ1, ν1 ≥1c1

,

and hence ∣∣∣∣µ1

ν1− 1∣∣∣∣ ≤ c1ε,

∣∣∣∣ ν1

µ1− 1∣∣∣∣ ≤ c1ε.

For any Borel set A ⊂ Sd−1 we deduce that

µ(A)µ1

≤ ν1

µ1

(ν(Aε)

ν1+

ε

ν1

)≤ (1 + c1ε)

(ν(Aε)

ν1+

ε

ν1

)≤ ν(Aε)

ν1+ c2ε.

By symmetry, we find

dP

(µ

µ1,

ν

ν1

)≤ c2ε.

For a function f : Sd−1 → R we set

‖f‖L := supx6=y

|f(x)− f(y)|‖x− y‖

, (29)

‖f‖∞ := supx|f(x)|, ‖f‖BL := ‖f‖L + ‖f‖∞. (30)

It follows from the proof of Corollary 11.6.5 in [8] that∣∣∣∣∫ fd

(µ

µ1− ν

ν1

)∣∣∣∣ ≤ 2‖f‖BLdP

(µ

µ1,

ν

ν1

).

Thus, for any function f : Sd−1 → R with ‖f‖BL ≤ 1 we get∣∣∣∣∫ fd(µ− ν)∣∣∣∣ ≤ µ1

[∣∣∣∣∫ fd

(µ

µ1− ν

ν1

)∣∣∣∣+ ∣∣∣∣ 1ν1

− 1µ1

∣∣∣∣ ∣∣∣∣∫ f dν

∣∣∣∣]≤ µ1(2c2ε + c3ε) = c4ε.

12

We may assume that K ⊂ RBd, then ‖h(K, ·)‖BL ≤ 2R (cf. [23], Lemma 1.8.10).Therefore,

|V (K)− V1(L, K)| =∣∣∣∣1d∫

Sd−1h(K, u)(µ− ν)(du)

∣∣∣∣ ≤ 2R

dc4ε = c5ε,

similarly |V (L)− V1(K, L)| ≤ c5ε. These estimates correspond to the inequalities(7.2.6) in [23], and the proof can now be completed as the proof of Lemma 7.2.3and of Theorem 7.2.2 in [23]. Note that the latter proof gives δ(K, L+x) ≤ cε1/d ifε is smaller than a certain positive constant ε1 depending only on d, r, R; if ε ≥ ε1,then the same inequality is achieved by a suitable choice of c.

4 Stability results in the Brunn-Minkowski-Fireytheory

A basic notion of the Brunn-Minkowski theory is the vector addition of convexbodies, which corresponds to the addition of support functions,

h(K + L, ·) = h(K, ·) + h(L, ·).

For p ≥ 1, a p-sum of convex bodies K, L ∈ Kd0 (the set of convex bodies in Rd

with 0 as interior point) can be defined by

h(K +p L, ·) := [h(K, ·)p + h(L, ·)p]1/p,

since the right-hand side is again a support function. Such p-means of convexbodies were introduced by Firey [9]. Lutwak [19], [20] has extended large parts ofthe Brunn-Minkowski theory to this more general combination of convex bodies.In this Brunn-Minkowski-Firey theory, as it is now called, the role of the classicalsurface area measures Sm(K, ·), m = 0, . . . , d − 1 (see, e.g., Section 4.2 of [23]) isplayed by measures Sp,i(K, ·) on the sphere Sd−1 (i = 0, . . . , d− 1). The measureSp,i(K, ·) is absolutely continuous with respect to Sd−1−i(K, ·) and has a Radon-Nikodym derivative given by

dSp,i(K, ·)dSd−1−i(K, ·)

= h(K, ·)1−p.

(Note that in [19] the measure S1,i(K, ·) is, unfortunately, denoted by Si(K, ·)and not by Sd−1−i(K, ·), as usual.) Lutwak’s theory contains an analogue of theAleksandrov-Fenchel-Jessen theorem, Corollary (2.3) of [19]: Suppose K, L ∈ Kd

0

and 0 ≤ i < d. If d − i 6= p > 1 and Sp,i(K, ·) = Sp,i(L, ·), then K = L. In thefollowing, we obtain a stability version of this result. Again, we use an assumptionon the Prohorov distance of two measures, which is weaker than the correspondingassumption for the total variation distance of the measures.

13

By Kd0(r, R) we denote the set of convex bodies K ⊂ Rd which satisfy rBd ⊂

K ⊂ RBd, where 0 < r < R.

4.1 Theorem. Let p > 1 and 0 < r < R. Suppose that K, L ∈ Kd0(r, R),

i ∈ {0, . . . , d− 1}, d− i 6= p, and

dP (Sp,i(K, ·), Sp,i(L, ·)) ≤ ε (31)

with some ε ≥ 0. Then

δ(K, L) ≤ cεq/2 with q =1

(d + 1)2d−i−2,

where the constant c depends only on d, p, r, R.

Proof. In the following, c1, c2, . . . denote positive constants which depend onlyon d, p, r, R. In the subsequent estimations where such constants occur, we veryoften tacitly use the facts that rBd ⊂ K, L ⊂ RBd, and that mixed volumes aremonotone in each argument.

With K and L as in the theorem, we use the notations (all integrations areover the sphere Sd−1)

Wi(K) =1d

∫h(K, u)Sd−1−i(K, du),

Wi(K, L) =1d

∫h(L, u)Sd−1−i(K, du) = V (K[d− 1− i], L[1], Bd[i]),

Wp,i(K, L) =1d

∫h(L, u)pSp,i(K, du)

=1d

∫h(L, u)ph(K, u)1−pSd−1−i(K, du).

As in [23], p. 398, we write, for some fixed i ∈ {0, . . . , d−1} and for k ∈ {0, . . . , d−i},

V(k) := V (K[d− i− k], L[k], Bd[i]),

thus Wi(K) = V(0), Wi(K, L) = V(1), Wi(L) = V(d−i). With these notations,Lutwak’s [19] inequality (IIp) (p. 132; see also Theorem 1.2 in [19]) reads

Wp,i(K, L)d−i ≥ V d−i−p(0) V p

(d−i). (32)

Interchanging K and L, we get

Wp,i(L,K)d−i ≥ V d−i−p(d−i) V p

(0). (33)

14

Another inequality proved by Lutwak [19] (p. 137) states that

Wp,i(K, L) ≥ Wi(K, L)pWi(K)1−p = V p(1)V

1−p(0) . (34)

Using (31), we can estimate as in Section 3 and obtain∣∣V(0) −Wp,i(L,K)∣∣ =

1d

∣∣∣∣∫ h(K, u)p[Sp,i(K, du)− Sp,i(L, du)]∣∣∣∣

≤ c1‖h(K, ·)p‖BLε,

hence|V(0) −Wp,i(L,K)| ≤ c2ε (35)

and similarly|Wp,i(K, L)− V(d−i)| ≤ c3ε. (36)

We write

Wp,i(K, L)− Vd−i−p

d−i

(0) Vp

d−i

(d−i)

=(

V(d−i)

V(0)

) pd−i(

Vd−i−p

d−i

(d−i) Vp

d−i

(0) −Wp,i(L,K))

+(

V(d−i)

V(0)

) pd−i [

Wp,i(L,K)− V(0)

]+[Wp,i(K, L)− V(d−i)

].

By (33), the first term on the right is not positive, hence (35) and (36) give

Wp,i(K, L)− Vd−i−p

d−i

(0) Vp

d−i

(d−i) ≤ c4ε. (37)

Now we assume that i ∈ {0, . . . , d− 2}. We write (37) in the form[Wp,i(K, L)− V p

(1)V1−p(0)

]+[V p

(1) − Vp(d−1−i)

d−i

(0) Vp

d−i

(d−i)

]V 1−p

(0) ≤ c4ε.

Here both brackets are nonnegative, the first by (34), and the second by theAleksandrov-Fenchel inequalities. We deduce that

Wp,i(K, L)− V p(1)V

1−p(0) ≤ c4ε (38)

andV p

(1) ≤ Vp(d−1−i)

d−i

(0) Vp

d−i

(d−i) + c5ε. (39)

Interchanging K and L in (39) gives

V p(d−1−i) ≤ V

p(d−1−i)d−i

(d−i) Vp

d−i

(0) + c5ε. (40)

15

Multiplication of (39) and (40) yields

V(1)V(d−1−i) ≤ V(0)V(d−i) + c6ε. (41)

We are now in the same situation as in the proof of Theorem 7.2.6 in [23]: theinequality there before (7.2.12) is precisely (41), with m replaced by d− i and c2

replaced by c6. Hence, the subsequent arguments in [23] (see the Appendix of thepresent paper) lead to the conclusion that

δ(K,L) ≤ c7εq (42)

(see also the hint at the end of the proof of Theorem 3.1). Here K = [K −s(K)]/b(K), where s(K) is the Steiner point and b(K) is the mean width of K.

We put λ = b(K)/b(L) and t = s(K)− λs(L), then (42) implies

δ(K, λL + t) ≤ c8εq. (43)

To derive (34), Holder’s inequality was used. In order to estimate t, we need asharper version of that inequality. We use a special case of an inequality by Kober[18], namely

w1a1 + w2a2 − aw11 aw2

2 ≥ w(a1/21 − a

1/22 )2

for a1, a2 ≥ 0 and w1, w2 > 0 with w1 + w2 = 1, where w := min{w1, w2}. Herewe put, for p > 1 and a, b > 0,

w1 =1p, w2 =

p− 1p

, a1 = apb1−p, a2 = b

and obtainapb1−p + (p− 1)b− pa ≥ mb1−p(ap/2 − bp/2)2 (44)

with m = min{1, p− 1}.Write h(M, ·) = hM for M ∈ Kd and put

I(M) :=1d

∫hM (u) Sd−1−i(K, du).

We apply (44) with

a =hL(u)I(L)

, b =hK(u)I(K)

,

where u ∈ Sd−1, and integrate over all u ∈ Sd−1 with respect to the measure(1/d)Sd−1−i(K, ·). The result can be written as

Wp,i(K, L)V p

(1)V1−p(0)

− 1 ≥ c9

∫ [(hL

I(L)

)p/2

−(

hK

I(K)

)p/2]2

h1−pK dSd−1−i(K, ·). (45)

16

The quotient hL/I(L) is invariant under a dilatation of L, hence on the right-handside, the body L can be replaced by λL. Therefore, (38) and (45) yield∫ [(

hλL

I(λL)

)p/2

−(

hK

I(K)

)p/2]2

dSd−1−i(K, ·) ≤ c10ε. (46)

Since I(M) is invariant under translations of M , inequality (43) shows that

|I(λL)− I(K)| ≤ c11εq.

Using this inequality and the mean value theorem, we can estimate, for u ∈ Sd−1,

|hλL(u)− hK(u)| ≤ c12

∣∣∣hλL(u)p/2 − hK(u)p/2∣∣∣

≤ c13

∣∣∣∣∣(

hλL(u)I(λL)

)p/2

−(

hK(u)I(K)

)p/2∣∣∣∣∣+ c14ε

q.

Together with (46), this yields an estimate(∫|hλL − hK | dSd−1−i(K, ·)

)2

≤ c15

∫|hλL − hK |2 dSd−1−i(K, ·) ≤ c16ε

q.

Since hλL+t(u) = hλL(u) + 〈u, t〉, it follows from (43) that

|〈u, t〉| ≤ |hλL(u)− hK(u)|+ c8εq

for u ∈ Sd−1. Writing t1 = t/‖t‖ if t 6= 0, we deduce that

‖t‖∫|〈u, t1〉|Sd−1−i(K, du) ≤ c17ε

q/2.

Now ∫|〈u, t1〉|Sd−1−i(K, du) ≥ c18,

since the integral is, up to a factor depending only on d, an intrinsic volume of aprojection of K and hence can be estimated from below by a constant dependingonly on d and r. The conclusion is that

‖t‖ ≤ c19εq/2. (47)

To estimate λ, we first deduce from (36) and (37) that

V(d−i) − Vd−i−p

d−i

(0) Vp

d−i

(d−i) ≤ c20ε,

17

thusV

d−i−pd−i

(d−i) − Vd−i−p

d−i

(0) ≤ c21ε.

Since we have assumed d− i− p 6= 0, this implies

Wi(L)−Wi(K) ≤ c22ε.

From (43), we get

K ⊂ λL + t + c8εqBd ⊂ (1 + c23ε

q)λL + t,

hence

Wi(L) ≤ Wi(K) + c22ε

≤ Wi((1 + c23εq)λL) + c22ε

= [(1 + c23εq)λ]d−iWi(L) + c22ε.

This gives λ ≥ 1− c24εq. Interchanging the roles of K and L, we similarly obtain

λ−1 ≥ 1− c25εq and hence

|λ− 1| ≤ c26εq. (48)

The inequalities (43), (47) and (48) finally give

δ(K, L) ≤ c27εq/2.

This completes the proof of Theorem 4.1 in the case where i ∈ {0, . . . , d− 2}.Finally, we consider the (simpler) case i = d − 1. As before, we deduce that

Wd−1(L)−Wd−1(K) ≤ c22ε, and hence by symmetry

|λ− 1| ≤ c28ε, (49)

where λ = b(K)/b(L). Using (37) and (45), we find that∫ [(hL

Wd−1(L)

)p/2

−(

hK

Wd−1(K)

)p/2]2

dσ ≤ c29ε

and thus ∫ ∣∣∣hp/2λL − h

p/2K

∣∣∣2 dσ ≤ c30ε.

An application of the mean value theorem shows that∫|hλL − hK |2 dσ ≤ c31ε,

hence (49) and Corollary 1 in [25] (see also Lemma 6.6.4 in [23]) give δ(K, L) ≤c32ε

1d+1 . The proof of Theorem 4.1 is now complete.

18

We remark that the preceding proof also permits us to give stability versions oftwo inequalities due to Lutwak [19]. The first of these is his inequality (IIp) (whichis (32) above).

4.2 Corollary. Let p > 1 and 0 < r < R. Suppose that K, L ∈ Kd0(r, R),

i ∈ {0, . . . , d− 1} and

Wp,i(K, L)−Wi(K)d−i−p

d−i Wi(L)p

d−i ≤ ε (50)

with some ε ≥ 0. Then there is a constant c depending only on d, p, r, R such that

δ(K, λL) ≤ cεq/2,

where λ = b(K)/b(L) and q is as in Theorem 4.1.

Proof. Assume that i ∈ {0, . . . , d − 2}. Then the assumption (50) implies thatδ(K,L) ≤ c33ε

q, by the argument after equation (37). The subsequent argumentin the proof of Theorem 4.1, which shows that |s(K)− λs(L)| ≤ c34ε

q/2, remainsthe same, hence δ(K, λL) ≤ c35ε

q/2, as stated.The case i = d− 1 can be treated as in the proof of Theorem 4.1.

The next result gives a stability version of Lutwak’s Corollary (1.3) (using hisnotations).

4.3 Corollary. Let p > 1, 0 < r < R and ϑ ∈ (0, 1). Suppose that K, L ∈Kd

0(r, R), i ∈ {0, . . . , d− 1} and

Wi((1− ϑ) ·K +p ϑ · L)p

d−i − (1− ϑ)Wi(K)p

d−i − ϑWi(L)p

d−i ≤ ε (51)

with some ε ≥ 0. Then there is a constant c depending only on d, p, r, R such that

δ(K, τL) ≤ cmin{ϑ, 1− ϑ}−q/2εq/2,

where τ is a suitable positive constant and q is as in Theorem 4.1.

Proof. Put M := (1 − ϑ) ·K +p ϑ · L. From the definitions of p-sums and of thefunctionals Wp,i, we obtain

Wi(M) = Wp,i(M, (1− ϑ) ·K +p ϑ · L)

= (1− ϑ)Wp,i(M,K) + ϑWp,i(M,L).

Since M ∈ Kd0(r, R), we can apply Corollary 4.2 and deduce that, with suitable

numbers τ1, τ2 > 0,

Wi(M) ≥ (1− ϑ)[c36δ(M, τ1K)

2q + Wi(M)

d−i−pd−i Wi(K)

pd−i

]+ϑ[c37δ(M, τ2L)

2q + Wi(M)

d−i−pd−i Wi(L)

pd−i

].

19

From this we infer that

Wi(M)p

d−i − (1− ϑ)Wi(K)p

d−i − ϑWi(L)p

d−i

≥ (1− ϑ)c38δ(M, τ1K)2q + ϑc39δ(M, τ2L)

2q

≥ min{ϑ, 1− ϑ}c40 [δ(M, τ1K) + δ(M, τ2L)]2q

≥ min{ϑ, 1− ϑ}c41δ(τ1K, τ2L)2q ≥ min{ϑ, 1− ϑ}c42δ(K, τL)

2q .

5 Stability of inverse integral transforms

The starting point of this section is formula (6),

Vd−1(Ku) =12

∫Sd−1

|〈u, v〉|Sd−1(K, dv), u ∈ Sd−1,

which expresses the projection function u 7→ Vd−1(Ku) of a convex body K as thecosine transform of its area measure of order d−1. The stability result of Bourgain& Lindenstrauss [2] is a quantitative version of the fact that two d-dimensionalconvex bodies with the same centre of symmetry must be close if their projectionfunctions are close. Groemer’s [14] book contains a detailed presentation of thistheorem and its proof (Theorem 5.5.7). In the present section, we use the methodof Bourgain and Lindenstrauss to obtain stability estimates for the inversion offurther integral transforms of area measures occurring in the geometry of convexbodies.

These integral transforms are of the following type. Let Φ : [−1, 1] → R be abounded, Borel measurable function. For a finite signed Borel measure µ on Sd−1,let

(TΦµ)(u) :=∫

Sd−1Φ(〈u, v〉)µ(dv) for u ∈ Sd−1. (52)

For a bounded measurable function f on Sd−1, the transform TΦf is defined asTΦµ for the signed measure µ = fσ, where σ denotes spherical Lebesgue measure.

We need a few facts about spherical harmonics, which can all be found in [14].If Yn is a spherical harmonic of degree n on Sd−1, then

TΦYn = ad,n(Φ)Yn

with

ad,n(Φ) = ωd−1

∫ 1

−1

Φ(t)P dn(t)(1− t2)(d−3)/2 dt,

where P dn is the Legendre polynomial of dimension d and degree n (e.g., [14], Th.

3.4.1). Here ωk = kκk is the area of the k-dimensional unit ball, and κk is itsvolume. The numbers ad,n(Φ) are called the multipliers of TΦ.

20

For f, g ∈ L2(Sd−1), the space of square integrable real functions on Sd−1, ascalar product is defined by

(f, g) :=∫

Sd−1fg dσ,

and the L2-norm by ‖f‖ :=√

(f, f). Let {Ynj : j = 1, . . . , N(d, n)} be an or-thonormal basis of the real vector space of spherical harmonics of degree n ∈ N0.For f ∈ L2(Sd−1), the relation

f ∼∞∑

n=0

Yn (53)

means that

Yn =N(d,n)∑

j=1

(f, Ynj)Ynj ,

and the series in (53) is called the condensed harmonic expansion of f ([14], p.72). Similarly, for a finite signed measure µ on Sd−1 we write

µ ∼∞∑

n=0

Yn (54)

if

Yn =N(d,n)∑

j=1

(∫Sd−1

Ynj dµ

)Ynj .

If (54) holds, then

TΦµ ∼∞∑

n=0

ad,n(Φ)Yn. (55)

The following theorem is only a slight extension of the result of Bourgain andLindenstrauss, to general transformations TΦ. For the reader’s convenience, werepeat the essential steps of the proof, in a simplified form, to indicate wherechanges are necessary. Recall that the norm ‖ · ‖BL was defined by (30) and that‖µ‖TV denotes the total variation norm of the signed measure µ.

5.1 Theorem. Assume that the multipliers of the transformation TΦ satisfy

ad,0(Φ) 6= 0, |ad,n(Φ)−1| ≤ bnβ for n ∈ N (56)

with suitable b, β > 0. Let µ be a finite signed measure on Sd−1, and let F :Sd−1 → R be a Lipschitz function. Then for each α ∈ (0, 1/(1 + β)) there is aconstant c depending only on d,Φ, α such that∣∣∣∣∫

Sd−1F dµ

∣∣∣∣ ≤ c‖F‖BL ‖µ‖1−αTV ‖TΦµ‖α.

21

If µ is even and (56) holds for even n, then the same conclusion can be drawn.

Proof. We choose b, β (depending on Φ) so that (56) holds. The constants c1, c2, . . .in the following depend only on d,Φ, b, β, α and hence only on d, Φ, α.

It was the idea of Bourgain and Lindenstrauss [2] to use the Poisson transform

µτ :=1ωd

∫Sd−1

1− τ2

(1 + τ2 − 2τ〈u, v〉)d/2µ(dv), u ∈ Sd−1,

for 0 < τ < 1. We have (all integrations are over Sd−1)∫Fτ dµ =

∫Fµτ dσ

and ∣∣∣∣∫ F dµ

∣∣∣∣ ≤∣∣∣∣∫ (F − Fτ ) dµ

∣∣∣∣+ ∣∣∣∣∫ Fτ dµ

∣∣∣∣≤ ‖F − Fτ‖∞ ‖µ‖TV +

∣∣∣∣∫ Fµτ dσ

∣∣∣∣ . (57)

For τ ≥ 1/4,

‖F − Fτ‖∞ ≤ 2d+1 ωd−1

ωd‖F‖L(1− τ) log

21− τ

(58)

([14], Lemma 5.5.8). Moreover,∣∣∣∣∫ Fµτ dσ

∣∣∣∣ ≤ ‖F‖ ‖µτ‖. (59)

If (54) is the condensed harmonic expansion of µ, then

µτ ∼∞∑

n=0

τnYn.

The maximal value of the function g(x) = xβτx for x > 0 is (−β/e log τ)β , hence

nβτn(1− τ)β ≤(

β

e

)β ( 1− τ

− log τ

)β

≤(

β

e

)β

for n ∈ N. Therefore, (56) gives

τn ≤ c1(1− τ)−β |ad,n(Φ)|.

Together with Parseval’s relation, this yields

‖µτ‖2 =∞∑

n=0

τ2n‖Yn‖2 ≤ c21(1− τ)−2β

∞∑n=0

|ad,n(Φ)|2‖Yn‖2.

22

Now (55) shows that‖µτ‖ ≤ c1(1− τ)−β‖TΦµ‖. (60)

From (57), (58), (59), (60) we get∣∣∣∣∫ F dµ

∣∣∣∣ ≤ c2‖F‖ ‖TΦµ‖(1− τ)−β + c3‖F‖L‖µ‖TV (1− τ) log2

1− τ

≤ c4‖F‖BL

[‖TΦµ‖(1− τ)−β + ‖µ‖TV (1− τ) log

21− τ

].

Since Φ is bounded, we have

‖TΦµ‖ ≤ c5‖µ‖TV .

Therefore, we can find a constant c6 and a number τ ∈ [ 14 , 1) such that

‖TΦµ‖(1− τ)−β = c6‖µ‖TV (1− τ) log2

1− τ.

For this τ and for α ∈ (0, 1) we get∣∣∣∣∫ F dµ

∣∣∣∣ ≤ c7‖F‖BL(1− τ)1−α(1+β)

(log

21− τ

)1−α

‖µ‖1−αTV ‖TΦµ‖α.

If now α < 1/(1 + β), then we get∣∣∣∣∫ F dµ

∣∣∣∣ ≤ c8‖F‖BL ‖µ‖1−αTV ‖TΦµ‖α.

If the signed measure µ is even, then the components in (54) satisfy Yn = 0for odd n. Therefore, one can conclude as above. This completes the proof ofTheorem 5.1.

The geometric applications are of the following type.

5.2 Theorem. Let Φ and β be as in Theorem 5.1, let 0 < r < R. Forγ ∈ (0, 1/d(1 + β)), there is a constant c depending only on d, Φ, γ, r, R with thefollowing property. If K, L ∈ Kd(r, R) and

µ := Sd−1(K, ·)− Sd−1(L, ·), (61)

thenδ(K, L + x) ≤ c‖TΦµ‖γ (62)

with a suitable vector x ∈ Rd.If K and L are centrally symmetric and (56) holds for even n, then the same

conclusion can be drawn.

23

Proof. The constants c1, c2, . . . in this proof depend only on d, Φ, γ, r, R. We applyTheorem 5.1 with α = dγ to the measure µ given by (61) and to F = hK , thesupport function of K. Without loss of generality, we assume that K ⊂ RBd.Since ‖µ‖TV = Sd−1(K, Sd−1) + Sd−1(L, Sd−1) can be estimated from above by aconstant depending only on R and d and the same is true for ‖hK‖BL (cf. [23],Lemma 1.8.10), we get ∣∣∣∣∫ F dµ

∣∣∣∣ ≤ c1‖TΦµ‖α.

By the geometric meaning of F and µ, this reads

|V (K)− V1(L,K)| ≤ c2‖TΦµ‖α,

and interchanging K and L we get

|V (L)− V1(K, L)| ≤ c2‖TΦµ‖α.

By a result of Diskant [4] (compare the remark at the end of the proof of Theorem3.1), the two inequalities

|V (K)− V1(L,K)| ≤ ε, |V (L)− V1(K, L)| ≤ ε

together implyδ(K, L + x) ≤ c3ε

1/d

for suitable x ∈ Rd, provided that ε ≤ ε0, where ε0 > 0 is a constant dependingonly on d, r, R. If c2‖TΦµ‖α ≤ ε0, then we get

δ(K, L + x) ≤ c4‖TΦµ‖α/d,

and if c2‖TΦµ‖α > ε0, the same estimate holds if c4 is chosen suitably.If K and L are centrally symmetric, then the signed measure µ is even. This

completes the proof of Theorem 5.2.

The special case of Theorems 5.1 and 5.2 treated by Bourgain and Lindenstraussconcerned the cosine transform, where Φ(t) = 1

2 |t| for t ∈ [−1, 1]. In that case, (56)holds for even n with β = (d+2)/2. Hence, for convex bodies K, L ∈ Kd(r, R) withthe same centre of symmetry and for the (d−1)st projection function Vd−1(K, u) =Vd−1(Ku), u ∈ Sd−1, one gets

δ(K, L) ≤ c‖Vd−1(K, ·)− Vd−1(L, ·)‖γ (63)

for γ ∈ (0, 2/d(d + 4)).It is natural to ask for similar results for the ith projection function,

Vi(K, u) = Vi(Ku) =12

∫Sd−1

|〈u, v〉|Si(K, dv), u ∈ Sd−1.

24

By a well-known integral geometric formula, the convex bodies K, L satisfyVi(K, ·) = Vi(L, ·) if the projections of K and L on an i-dimensional subspacealways have the same i-dimensional volume. For i = 1, a strong stability resultwas proved by Goodey and Groemer [13]. In two books, the question for corre-sponding generalizations was posed. Groemer [14], p. 222, writes that ‘at presentsuch stability estimates exist only in the cases i = 1 and i = d− 1’. Gardner [10]asks in his Problem 4.7 (p. 157) whether a stability result of the type (63) canbe obtained for 1 < i < d − 1. Curiously, a positive answer on the basis of pub-lished results could have been given at the time when those books were written.In fact, the analytic part of the Bourgain-Lindenstrauss [2] proof (just replace µby µi := Si(K, ·)− Si(L, ·) in the first part of the proof of Theorem 5.2) gives

|V(0) − V(i)| ≤ c2‖TΦµi‖α, |V(i+1) − V(1)| ≤ c2‖TΦµi‖α

for α ∈ (0, 2/(d + 4)) (if Φ(t) = 12 |t|), where

V(k) := V (K[i + 1− k], L[k], Bd[d− 1− i]).

As shown in [23] (Proof of Lemma 7.2.3), the inequalities

|V(0) − V(i)| ≤ ε, |V(i+1) − V(1)| ≤ ε

for K, L ∈ Kd(r, R) and some ε > 0 imply

0 ≤ V(1) − Vi/(i+1)(0) V

1/(i+1)(i+1) ≤

(R

r+ 1)

ε.

One can now essentially use the proof of a stability result for the Aleksandrov-Fenchel-Jessen theorem ([23], Theorem 7.2.6) to deduce the following.

5.3 Theorem. Let i ∈ {2, . . . , d − 2} and 0 < r < R, let K, L ∈ Kd(r, R) beconvex bodies which are centrally symmetric with the same centre. For

γ ∈(

0,1

(d + 1)(d + 4)2i−2

),

there exists a constant c depending only on d, γ, r, R such that

δ(K, L) ≤ c‖Vi(K, ·)− Vi(L, ·)‖γ .

We turn to other integral transforms of type TΦ which have occurred in geometriccontexts. The sine transform is the transformation TΦ with Φ(t) =

√1− t2 for

t ∈ [−1, 1]. If K ∈ Kd is a convex body and u ∈ Sd−1, then

V (d−1)(K, u) :=∫ ∞

−∞Vd−2(K ∩ (u⊥ + tu)) dt

=1

2(d + 1)

∫Sd−1

√1− 〈u, v〉2 Sd−1(K, dv) (64)

25

(see [21], p. 60); here 2Vd−2(K ′) is the (d−2)-dimensional surface area of a (d−1)-dimensional convex body K ′. Thus the functional V (d−1)(K, ·), the integratedsurface area of parallel hyperplane sections, is, up to a factor, the sine transformof the surface area measure of K. The sine transform S is connected with the cosinetransform C and the spherical Radon transform R by the relation RC = κd−2S,which is easily obtained by a direct calculation. (For convex bodies this implies,in view of (64), that Radon transforms of projection functions are connected withsections; this interplay was studied in greater generality by Goodey [11].) Thisrelation implies corresponding relations for the multipliers: if

f ∼∞∑

n=0

Yn, Cf ∼∞∑

n=0

ζd,nYn, Rf ∼∞∑

n=0

ρd,nYn,

then

κd−2Sf ∼∞∑

n=0

ρd,nζd,nYn.

For even n, we have |ζ−1d,n| = O

(n(d+2)/2

), as remarked above, and |ρ−1

d,n| =O(n(d−2)/2

)(as follows from [14], Lemma 3.4.7 and (3.4.19)), hence |ρ−1

d,nζ−1d,n| =

O(nd). Thus, for the function Φ(t) =√

1− t2, assumption (56) holds for even nwith β = d. This gives the following result.

5.4 Theorem. Let 0 < r < R, let K, L ∈ Kd(r, R) be convex bodies with the samecentre of symmetry. For γ ∈ (0, 1/d(d + 1)), there exists a constant c dependingonly on d, γ, r, R such that

δ(K, L) ≤ c‖V (d−1)(K, ·)− V (d−1)(L, ·)‖γ .

The results involving the cosine or sine transform are necessarily restricted tocentrally symmetric convex bodies, since a transform TΦµ with an even functionΦ does not contain information on the odd part of the signed measure µ. Weturn now to stability versions of some uniqueness theorems for not necessarilysymmetric convex bodies.

Anikonov and Stepanov [1] have proposed to consider, for K ∈ Kd and u ∈Sd−1, besides the projection volume Vd−1(K, u) = Vd−1(Ku), also the area S(K, u)of the illuminated portion of K in direction u, that is,

S(K, u) := Sd−1(K, {v ∈ Sd−1 : 〈u, v〉 ≥ 0}).

They showed that the combined functional

F (K, u) := pVd−1(K, u) + qS(K, u), u ∈ Sd−1,

with constants p, q (p, q, 2pκd−1+qωd 6= 0) determines the convex body K uniquelyup to a translation. They also proved a corresponding stability result in R3. This,

26

however, is rather weak, since it assumes that the difference F (K, ·) − F (L, ·) issmall in a norm that involves derivatives up to order six. A stronger result can beobtained with the aid of Theorem 5.2. In fact, we have F (K, ·) = TΦSd−1(K, ·)with Φ = pΦ1+qΦ2, where Φ1(t) = 1

2 |t| and Φ2 = 1[0,1]. Now, ad,n(Φ1) = 0 for oddn, and |ad,n(Φ1)−1| = O(n(d+2)/2) for even n. On the other hand, ad,n(Φ2) = 0for even n > 0, and |ad,n(Φ2)−1| = O(nd/2) for odd n (see [14], Lemma 3.4.6and (3.4.20)). It follows that Φ satisfies (56) with β = (d + 2)/2 (note that theassumption 2pκd−1 + qωd 6= 0 ensures that ad,0(Φ) 6= 0). Hence, for any twoconvex bodies K, L ∈ Kd(r, R), we have

δ(K, L + x) ≤ c‖F (K, ·)− F (L, ·)‖γ

with a suitable vector x ∈ Rd, for γ ∈ (0, 2/d(d + 4)) and with c depending onlyon d, p, q, γ, r, R.

The last two transformations to be considered stem from the part of theoreticalstereology or geometric tomography where one is interested in obtaining informa-tion on convex bodies from lower dimensional sections. The second mean sectionbody M2(K) of a convex body K ∈ Kd was introduced by Goodey and Weil [12].It is defined by

h(M2(K), ·) =∫Ed2

h(K ∩ E, ·) µ2(dE).

Here Ed2 is the affine Grassmannian of two-dimensional planes in Rd and µ2 is its

motion invariant measure, normalized so that µ2({E ∈ Ed2 : E ∩Bd 6= ∅}) = κd−2.

Thus, M2(K) comprises information about the two-dimensional sections of K, inintegrated form. Goodey and Weil showed that two d-dimensional convex bodiesK and L with M2(K) = M2(L) differ only by a translation, and they mentionedbriefly (on p. 429) that a corresponding stability version could be obtained. Wewill make this more explicit.

For unit vectors u, v, let α(u, v) ∈ [0, π] denote the angle between u and v.Goodey and Weil (loc. cit., Corollary 2) proved that

h(M2(K)− t, u) =κ2κd−2(

d2

)κd

∫Sd−1

α(u, v) sinα(u, v) Sd−1(−K, dv) (65)

for u ∈ Sd−1, where t is a suitable translation vector. Their proof (cf. Theorem2) provides no information about this translation, but a related remark of Goodey[11], p. 165, gives a hint. Denoting by zr+1(K) the intrinsic (r + 1)st momentvector of K (see [23], p. 304), we have

1κd

∫Sd−1

h(M2(K)− t, u)u σ(du) = z1(M2(K)− t) = z1(M2(K))− t

=κ2κd−2(

d2

)κd

zd−1(K)− t.

27

But this must be the zero vector, as we see by using (65), Fubini’s theorem, thefact that a vector integral of the form∫

Sd−1f(〈u, v〉) u σ(du)

is invariant under rotations fixing v and hence is a multiple of v, and that∫Sd−1 v Sd−1(−K, dv) = 0. We deduce that the body

M ′2(K) :=

(d2

)κd

κ2κd−2M2(K)− zd−1(K),

which we call the normalized second mean section body of K, satisfies

h(M ′2(K), u) =

∫Sd−1

α(u, v) sinα(u, v) Sd−1(−K, dv) for u ∈ Sd−1.

Thus, h(M ′2(K), ·) = TΦSd−1(−K, ·) with Φ(t) = (arccos t)

√1− t2 for t ∈ [−1, 1].

From a computation in [12] (formula (4.10), together with the relation

cn(t) =(

n + d− 3d− 3

)P d

n(t)

between Gegenbauer and Legendre polynomials, see [14], p. 97) it follows that

ad,n(Φ) =c(d)

(n− 1)(n + d− 1)

(n!Γ

(12 (n + d)

)(n + d− 2)!Γ

(12 (n + 2)

))2

,

with a constant c(d) depending only on d. From this we deduce that (56) holdswith β = d. For the resulting stability estimate, we may now use the Hausdorffdistance also on the right-hand side: For γ ∈ (0, d(d + 1)), there exists a constantc depending only on d, r, R, γ such that, for K, L ∈ Kd(r, R),

δ(K, L + x) ≤ cδ(M ′2(K),M ′

2(L))γ

for a suitable vector x ∈ Rd.

The origin of our last example is an investigation, [24], on the oriented mean normalmeasure of a stationary stochastic process of convex particles and its determinationfrom planar sections. There one has reason to consider the function defined by

V(d−1)+ (K, u) :=

∫ ∞

−∞Hd−2(∂uK ∩ (u⊥ + tu)) dt

for K ∈ Kd and u ∈ Sd−1; here Hd−2 denotes the (d − 2)-dimensional Hausdorffmeasure and ∂uK is the ‘upper boundary’ of K in direction u, that is, the set of

28

all boundary points of K at which there exists an outer unit normal vector v with〈u, v〉 ≥ 0. By formula (17) of [24],

V(d−1)+ (K, ·) = TΦSd−1(K, ·)

with Φ(t) =√

1− t21[0,1] (so that TΦ could be called the hemispherical sine trans-form). For even n ∈ N, the multipliers of TΦ are essentially those of the sinetransform, namely ad,n(Φ) = 1

2ad,n(Ψ) for Ψ(t) =√

1− t2. Hence, as shown be-fore Theorem 5.4, |ad,n(Φ)−1| = O(nd) for even n. For odd n, the multipliers havenot been determined explicitly, but it has been shown that ad,1(Φ) 6= 0 and, forodd n ≥ 3,

ad,n(Φ) = ωd−11 · 3 · · · (n− 2)

(d− 1)(d + 1) · · · (d + n− 4)f(n)

(n + d− 1)(n + d− 3)

withn + d− 3 < (−1)(n−1)/2f(n) ≤ n + d− 1

([24], p. 36 together with (20) and (19)). From this, one obtains |ad,n(Φ)−1| =O(nd/2) for odd n. We deduce that (56) holds with β = d. Hence, for 0 < r < Rand γ ∈ (0, 1/d(d + 1)) there exists a constant c depending only on d, γ, r, R suchthat, for K ∈ Kd(r, R),

δ(K, L + x) ≤ c‖V (d−1)+ (K, ·)− V

(d−1)+ (L, ·)‖γ

with a suitable vector x ∈ Rd.

6 Appendix

In the proofs of Theorems 4.1 and 5.3 we have referred to the proof of Theorem7.2.6 in [23], which in turn relies on inequality (6.4.9) of [23] (p. 335). The proofof (6.4.9) given there is not complete, as A. Giannopoulos has kindly pointed out.We take this opportunity to correct the error (using the same notations). Theproof of (6.4.9) as given is correct if U12U00 − U01U02 < 0; observe that

U201 − U00U11 ≥ 0, U2

02 − U00U22 ≥ 0. (66)

Now, for λ1, λ2 ≥ 0 also

0 ≤ V (λ1K1 + λ2K2,K0, C)2

−V (λ1K1 + λ2K2, λ1K1 + λ2K2, C)V (K0,K0, C)

= λ21(U

201 − U00U11) + λ2

2(U202 − U00U22)− 2λ1λ2(U12U00 − U01U02).

If U12U00 − U01U02 > 0, we can deduce (6.4.9) from this inequality. If U12U00 −U01U02 = 0, (6.4.9) holds by (66).

29

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Authors’ address:

Mathematisches InstitutAlbert-Ludwigs-UniversitatEckerstr. 1D-79104 Freiburg i.Br.Germany

daniel.hug@math.uni-freiburg.derschnei@uni-freiburg.de

31