STABILITY RESULTS INVOLVING SURFACE AREA MEASURES OF CONVEX BODIES DANIEL HUG AND ROLF SCHNEIDER

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1 STABILITY RESULTS INVOLVING SURFACE AREA MEASURES OF CONVEX BODIES DANIEL HUG AND ROLF SCHNEIDER Abstract We strengthen some known stability results from the Brunn-Minkowski theory an obtain new results of similar types. These results concern pairs of convex boies for which either surface area measures, or counterparts of such measures in the Brunn-Minkowski-Firey theory, or geometrically significant transforms of such measures, are close to each other. MSC 2000: 52A20, 52A40. 1 Introuction In recent ecaes, several of the classical uniqueness theorems for convex boies have been turne into quantitative versions, in the form of stability results. The starting point for the present investigation are uniqueness theorems of Minkowski an Aleksanrov, respectively. Minkowski s theorem, in its later general form, says that a -imensional convex boy is uniquely etermine, up to a translation, by its ( 1)st surface area measure. A theorem of Aleksanrov, inepenently prove by Fenchel an Jessen, states the extension of this result to lower orer surface area measures. Aleksanrov s projection theorem asserts that a -imensional convex boy with a given centre of symmetry is uniquely etermine by the volumes (or intrinsic volumes of a given positive orer) of its ( 1)-imensional orthogonal projections. Also this theorem involves surface area measures, since volumes of ( 1)-imensional orthogonal projections an area measures are relate by the cosine transform, see (6). In the following, we improve some known stability results corresponing to these uniqueness theorems, an we obtain new stability versions of some similar uniqueness assertions. To formulate a stability version of Minkowski s uniqueness theorem, we enote by K (r, R) the set of convex boies in Eucliean space R which contain some ball of raius r > 0 an are containe in some ball of raius R > r. Let K, L K (r, R) be convex boies whose surface area measures S 1 (K, ) an S 1 (L, ) satisfy S 1 (K, ) S 1 (L, ) ɛ. (1) A typical stability version of Minkowski s theorem requires to fin a number α > 0, which epens only on, an a number c > 0 epening only on, r, R such that for ɛ 0, inequality (1) implies δ(k, L + x) cɛ α (2) 1

2 with a suitable x R ; here δ enotes the Hausorff metric. The best result of this type up to now is ue to Diskant [4] (Theorem in [23]); it gives a stability orer α = 1/. Uner the stronger assumption that K, L are of class C 2 + an that S 1 (K, ) S 1 (L, ) ɛσ, (3) where σ enotes the spherical Lebesgue measure, Diskant [6] (with etails in [7]) obtaine (2) with the better exponent α = 1/( 1). Our first result, to be prove in Section 2, will achieve (2) with α = 1/( 1) uner the weaker assumption (1), for a large class of convex boies incluing polytopes an boies of class C 2 +. We also show that in this result the exponent 1/( 1) is optimal. Assumption (1) is essentially equivalent to an assumption on the total variation norm of the ifference of the surface area measures of K an L: (1) implies S 1 (K, ) S 1 (L, ) T V 2ɛ, (4) an (4) implies (1) with ɛ replace by 2ɛ. Following a suggestion of Wolfgang Weil, we replace this assumption on the total variation istance of measures by a more natural one on the Prohorov istance P. The inequality (1) implies P (S 1 (K, ), S 1 (L, )) ɛ, (5) but not conversely. We show in Section 3 that the weaker assumption (5) is still sufficient to obtain the stability estimate (2) for all convex boies K, L K (r, R) with α = 1/. For the Aleksanrov-Fenchel-Jessen theorem on lower orer surface area measures (Corollary in [23]), a stability version was obtaine by Schneier [22] (Theorem in [23]). Lutwak s work on the Brunn-Minkowski-Firey theory, where Minkowski sums of convex boies are replace by Firey s p-sums, contains also a generalization of the Aleksanrov-Fenchel-Jessen theorem (Corollary (2.3) in [19]). In Section 4 we will give a stability result for this extene theorem. As corollaries of the proof, we obtain stability versions of two inequalities of Lutwak [19]. For a convex boy K R an a unit vector u S 1, we enote by K u the image of K uner orthogonal projection onto u, the hyperplane through 0 orthogonal to u. We write V 1 ( ) for the volume in ( 1)-imensional hyperplanes. Then V 1 (K u ) = 1 u, v S 1 (K, v), (6) 2 S 1 where, is the scalar prouct of R. Thus, the projection function u V 1 (K u ) of K is, up to a constant factor, the cosine transform of the surface area measure S 1 (K, ). A special case of Aleksanrov s projection theorem (e.g., Theorem in [10]) says that two -imensional centrally symmetric convex boies with the same projection function iffer only by a translation. Stability 2

3 versions of this uniqueness theorem are ue to Campi [3] (for = 3) an to Bourgain & Linenstrauss [2]. In Section 5 we use the metho of Bourgain an Linenstrauss to obtain further stability results of a similar nature. One of these results concerns the sine transform u 1 u, v 2 S 1 (K, v) S 1 of the surface area measure, which also has geometric significance. Then we obtain some stability results for convex boies which are not necessarily centrally symmetric. They refer to various integral transforms, appearing in work of Anikonov & Stepanov [1], Gooey & Weil [12], Schneier [24]. 2 Stability for Minkowski s theorem We work in -imensional real vector space R ( 3), equippe with the stanar Eucliean structure. The set of convex boies (non-empty compact convex sets) in R is enote by K. For notions from the theory of convex boies which are not explaine here, we refer to [23]. Apart from replacing E n by R, we use the terminology of that book. Let K K be a convex boy, an let S 1 (K, ) be its surface area measure of orer 1. It is a finite Borel measure on the unit sphere S 1. By Lebesgue s ecomposition theorem, it can be ecompose, with respect to the ( 1)-imensional Hausorff measure H 1, into an absolutely continuous part S 1 a (K, ) an a singular part Ss 1 (K, ). The latter can be ecompose further, by efining S 1(K, c ) := S 1 (K, {u})δ u, (7) u S 1 where δ u enotes the Dirac measure (unit point mass) at u, an S n 1(K, ) := S s 1(K, ) S c 1(K, ). Clearly, in (7) at most countably many summans are non-zero. Moreover, S 1 (K, {u}) = H 1 (F (K, u)) for u S 1, where F (K, u) is the support set of K with outer normal vector u. Thus, we have the ecomposition S 1 (K, ) = S a 1(K, ) + S c 1(K, ) + S n 1(K, ) (8) of the surface area measure S 1 (K, ) into an absolutely continuous measure S 1 a (K, ), a component Sc 1 (K, ) which is an at most countable sum of point masses, an a singular component S 1 n (K, ) without point masses. 2.1 Theorem. Let 0 < r < R. There exists a number c, which epens only on, r, R, with the following property. If K, L K (r, R) are convex boies satisfying 3

4 S 1 n (K, ) = 0, Sn 1 (L, ) = 0 an the assumption S 1 (K, ) S 1 (L, ) ɛ (9) for some ɛ 0, then for a suitable vector x R. δ(k, L + x) cɛ 1 1 The conition S 1 n (K, ) = 0 is fulfille, for example, if K is a polytope, or if the surface area measure of K is absolutely continuous. The latter is true, in particular, if the support function of K is of class C 2. The exponent 1/( 1) in Theorem 2.1 is optimal, at least for the class of polytopes. This can be seen by choosing for K a unit cube an for L the polytope which is obtaine from K by cutting off a vertex of K in such a way that the section plane meets K in a regular ( 1)-simplex of ege length ɛ 1/( 1). On the other han, uner the stronger assumption that L is a ball an that (3) hols, 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] an makes also use of Diskant [5]. The proof oes not require the full conition S 1 n (K, ) = 0 = Sn 1 (L, ), but only its consequence S n 1((1 t)k + tl, ) max{s n 1(K, ), S n 1(L, )} for 0 t 1; (10) hence we will work uner this assumption. As usual, we write V 1 (K, L) := V (K[ 1], L) for K, L K, where V enotes the mixe volume, thus V 1 (K, L) = 1 h(l, u)s 1 (K, u). S 1 Here h(l, ) is the support function of L. We also write V ( ) for the volume functional in R. We assume that K, L K (r, R) are convex boies satisfying (9) an (10). For t [0, 1] we set H t := (1 t)k + tl, then H t K (r, R). In the following, c 1, c 2,... enote positive constants which epen only on, r, R. The proof is ivie into four steps. Step I. First we show that V (H t ) V (K) c 1 ɛ, V (H t ) V (L) c 1 ɛ (11) 4

5 for t [0, 1]. By Lemma in [23], the estimates V (L) V 1 (K, L) c 2 ɛ, (12) 0 V 1 (K, L) V (K) 1 V (L) 1 c3 ɛ (13) an the corresponing ones with K an L interchange follow from the assumptions on K an L. From V 1 (K, L) V (K) 1 V (L) an (12) we euce that hence ( V (L) 1 + c ) 2ɛ = (V (L) + c 2 ɛ) V (K) 1 V (L), V (L) By symmetry, we infer that V (K) V (L) (1 + c 4 ɛ) 1 V (L) + c 5 ɛ. V (K) V (L) c 5 ɛ. Since K, L K (r, R), this implies V (K) 1/ V (L) 1/ c6 ɛ. (14) The function φ efine by φ(t) := V (H t ) 1/ (1 t)v (K) 1/ tv (L) 1/, t [0, 1], is concave an satisfies φ(0) = φ(1) = 0, hence Using (13) an K K (r, R) we get φ (0) φ(t) 0 for t [0, 1]. (15) φ (0) = V 1(K, L) V (K) 1 V (L) 1 V (K) 1 c 7 ɛ. (16) Hence, by (14), (15) an (16) we obtain V (H t ) 1/ V (K) 1/ V (L) 1/ V (K) 1/ + c7 ɛ c 8 ɛ. Since H t, K K (r, R), this implies the first inequality of (11), an the secon follows by symmetry. Step II. Next we show an analogue of (11) for surface areas, namely V1 (H t, B ) V 1 (K, B ) c9 ɛ, V1 (H t, B ) V 1 (L, B ) c9 ɛ (17) 5

6 for t [0, 1]; here B is the unit ball. If the support function h(m, ) of the convex boy M K is secon orer ifferentiable at u S 1 (which hols for H 1 almost all u S 1 ), then the eigenvalues of the secon orer ifferential 2 h(m, ) u at u are the principal raii of curvature of M at the point with outer normal vector u. Their prouct is enote by D 1 h(m, u), thus D 1 h(m, u) = et( 2 h(m, u) u ). For all u S 1 with the property that h(k, ) an h(l, ) are secon orer ifferentiable at u, an thus for H 1 almost all u S 1, we set m(u) := min{d 1 h(k, u), D 1 h(l, u)}. Minkowski s eterminant inequality states that hence D 1 h(h t, u) 1 1 (1 t)d 1 h(k, u) td 1 h(l, u) 1 1, D 1 h(h t, u) m(u) (18) for all t [0, 1]. Let ω 1 enote the measurable set of all u S 1 such that h(k, ) an h(l, ) are secon orer ifferentiable at u an D 1 h(k, u) > m(u). Then m(u) = D 1 h(l, u) for u ω 1 an 0 (D 1 h(k, u) m(u))h 1 (u) S 1 = (D 1 h(k, u) D 1 h(l, u))h 1 (u) ω 1 = S 1 (K, ω 1 ) S 1 (L, ω 1 ). Here we have use the fact that if M K an ω is the set of all u S 1 such that h(m, ) is secon orer ifferentiable at u, then the restriction of the measure S 1 (M, ) to ω is absolutely continuous with respect to H 1. This can be verifie on the basis of [15], [16] (an with the terminology use there): Choose any normal bounary point x τ(m, ω) of M, that is, x = τ M (u) for a uniquely etermine u ω. Then by an inspection of the proofs of Lemma 3.4 an Lemma 3.1 in [15], we obtain k i (x, u) > 0 for i {1,..., 1}; moreover, by Lemma 3.1 in [15], we fin k i (x) = k i (x, u) for i {1,..., 1}. This shows that H 1 (M, x) > 0 for H 1 almost all x τ(m, ω). The assertion is then implie by Theorem 3.7 in [16]. Now it follows from (9) that 0 (D 1 h(k, u) m(u))h 1 (u) ɛ. (19) S 1 6

7 an For M K an u S 1, we set f(m, u) := V 1 (F (M, u)) m(u) := min{f(k, u), f(l, u)}. The aitivity of support sets ([23], Theorem 1.7.5(c)), together with the Brunn- Minkowski theorem, implies that an therefore f(h t, u) 1 1 (1 t)f(k, u) tf(l, u) 1 1 f(h t, u) m(u) (20) for u S 1 an all t [0, 1]. Let ω 2 enote the set of all u S 1 such that f(k, u) > m(u). Hence, ω 2 is at most countable, an for u ω 2 we have m(u) = f(l, u) an f(k, u) m(u) = S 1 (K, {u}) S 1 (L, {u}), thus 0 (f(k, u) m(u)) = (f(k, u) m(u)) u S 1 u ω 2 = S 1 (K, ω 2 ) S 1 (L, ω 2 ). Therefore, (9) implies 0 u S 1 (f(k, u) m(u)) ɛ. (21) For convex boies M, H K, we now make use of the ecomposition V 1 (M, H) = 1 h(h, u) [ S 1(M, a u) + S 1(M, c u) + S 1(M, n u) ] S 1 = V a 1 (M, H) + V c 1 (M, H) + V n 1 (M, H) with V1 a (M, H) := 1 h(h, u)d 1 h(m, u)h 1 (u), V c 1 (M, H) := 1 h(h, u)f(m, u), V1 n (M, H) := 1 h(h, u)s n 1(M, u). 7

8 Here, as below, we write instea of if the integration is extene over S 1. S 1 Similarly, means u S 1, where the summation effectively extens only over countably many summans. We estimate the expression I := V (H t ) V1 n (K, H t ) 1 h(h t, u)m(u)h 1 (u) 1 h(ht, u)m(u) from both sies. Without loss of generality, we assume that rb K, L. Then the support function of H t satisfies r h(h t, ) 2R. First we insert in I the expression V n 1 (K, H t ) = V 1 (K, H t ) + V a 1 (K, H t ) + V c 1 (K, H t ) an use H t = (1 t)k + tl together with the estimates (11), (19), (21) to obtain I = V (H t ) 1 h(h t, u)s 1 (K, u) + 1 h(h t, u)(d 1 h(k, u) m(u))h 1 (u) + 1 h(ht, u)(f(k, u) m(u))h 1 (u) (1 t)(v (H t ) V (K)) + t(v (H t ) V 1 (K, L)) + 2R [ (D 1 h(k, u) m(u))h 1 (u) + ] (f(k, u) m(u)) t(v (H t ) V 1 (K, L)) + c 10 ɛ. (22) Next, we insert in I the ecomposition V (H t ) = V a 1 (H t, H t ) + V c 1 (H t, H t ) + V n 1 (H t, H t ) an use the fact that S 1 n (H t, ) S 1 n (K, ) is, by (10), a positive measure. Together with (18) an (20), this gives I = 1 h(h t, u) ( S n 1(H t, u) S 1(K, n u) ) + 1 h(h t, u) (D 1 h(h t, u) m(u)) H 1 (u) + 1 h(ht, u)(f(h t, u) m(u)) r { S n 1 ( Ht, S 1) S n 1(K, S 1 ) 8

9 + (D 1 h(h t, u) m(u))h 1 (u) + (f(h t, u) m(u)) } r { S n 1 (H t, S 1 ) S n 1(K, S 1 ) + (D 1 h(h t, u) D 1 h(k, u))h 1 (u) + (f(h t, u) f(k, u)) } = r(v 1 (H t, B ) V 1 (K, B )). (23) Combining (22) an (23), we fin By (11) an (12), an thus V 1 (H t, B ) V 1 (K, B ) t r (V (H t) V 1 (K, L)) + c 11 ɛ. V (H t ) V 1 (K, L) V (H t ) V (L) + V (L) V 1 (K, L) c 12 ɛ V 1 (H t, B ) V 1 (K, B ) c 13 ɛ, t [0, 1]. (24) On the other han, from (18), (20), (10), (19), (21) we euce V 1 (H t, B ) = V1 a (H t, B ) + V1 c (H t, B ) + V1 n (H t, B ) 1 m(u)h 1 (u) m(u) + Sn 1(H t, S 1 ) = 1 D 1 h(k, u)h 1 (u) f(k, u) + Sn 1(K, S 1 ) 1 (D 1 h(k, u) m(u))h 1 (u) 1 1 ( ( (f(k, u) m(u)) + S n 1 Ht, S 1) S 1(K, n S 1 ) ) V 1 (K, B ) 2 ɛ. (25) Now (24) an (25) yiel the first estimate of (17), an the secon follows by interchanging K an L. The next step provies corresponing estimates for projection volumes. Step III. If v S 1 an t [0, 1], then V 1 (H v t ) V 1 (K v ) c 14 ɛ, V 1 (H v t ) V 1 (L v ) c 14 ɛ. (26) 9

10 For the proof, we efine J := V 1 (Ht v ) 1 u, v S n 2 1(K, u) 1 u, v m(u)h 1 (u) 1 u, v m(u). 2 2 Since J = 1 2 u, v (D 1 h(h t, u) m(u))h 1 (u) + 1 u, v (f(ht, u) m(u)) u, v ( S n 2 1(H t, u) S 1(K, n u) ) 0, we can euce that V 1 (H v t ) V 1 (K v ) 1 2 u, v (D 1 h(k, u) m(u))h 1 (u) 1 u, v (f(k, u) m(u)) 2 ɛ. (27) On the other han, by (17), (19), (21) V 1 (Ht v ) V 1 (K v ) J = 1 2 u, v ( S 1(H n t, u) S 1(K, n u) ) u, v (D 1 h(h t, u) m(u)) H 1 (u) + 1 u, v (f(ht, u) m(u)) 2 S 1(H n t, S 1 ) S 1(K, n S 1 ) + (D 1 h(h t, u) m(u))h 1 (u) + (f(h t, u) m(u)) = (V 1 (H t, B ) V 1 (K, B )) 10

11 + (D 1 h(k, u) m(u))h 1 (u) + (f(k, u) m(u)) c 15 ɛ. (28) The estimates (27) an (28) yiel the first estimate in (26), an the secon estimate follows by symmetry. Step IV. The rest of the proof now follows from the work of Diskant [5], [7]. For given v S 1, the function efine by φ v (K, L, t) := V 1 (H v t ) 1 1 (1 t)v 1 (K v ) 1 1 tv 1 (L v ) 1 1 for t [0, 1] can be estimate, in view of (26), by ] φ v (K, L, t) = (1 t) [V 1 (Ht v ) 1 1 V 1 (K v ) 1 1 +t [V 1 (Ht v ) 1 1 V 1 (L v ) 1] c 16 ɛ. If λ > 0 is such that V 1 (λk v ) = V 1 (L v ), one obtains φ v (λk, L, t) c 17 ɛ. Now the main theorem of [5] shows that there exist ɛ 0 > 0 an c 18, epening only on, r, R, such that δ(λk v, L v + x(v)) c 18 ɛ 1 1 for some x(v) v, if ɛ ɛ 0, an therefore δ(k v, L v + x(v)) c 19 ɛ 1 1. For ɛ > ɛ 0, the same inequality hols if the constant c 19 is ajuste. Thus the assertion of Theorem 2.1 is implie by Theorem in [10]. 3 Stability an Prohorov metric For a set A S 1 an for ɛ > 0, let A ɛ := {y S 1 : x y < ɛ for some x A}, where is the Eucliean norm. For finite Borel measures µ, ν on S 1, let P (µ, ν) := inf { ɛ > 0 : µ(a) ν(a ɛ ) + ɛ an ν(a) µ(a ɛ ) + ɛ for all Borel sets A S 1}. 11

12 This efines the Prohorov metric P, 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 in [23]), replacing the assumption (1) by the weaker assumption (5). 3.1 Theorem. Let 0 < r < R. There exists a number c, epening only on, r, R, such that, for K, L K (r, R), for some x R. δ(k, L + x) c P (S 1 (K, ), S 1 (L, )) 1/ Proof. In the following, the positive constants c 1, c 2,... epen only on, r, R. Let K, L K (r, R). We set µ := S 1 (K, ), ν := S 1 (L, ), µ 1 := µ(s 1 ), ν 1 := ν(s 1 ), ɛ := P (µ, ν). Then an hence µ 1 1 ν 1 c 1ɛ, µ 1 ν 1 ɛ, µ 1, ν 1 1 c 1, ν 1 1 µ 1 c 1ɛ. For any Borel set A S 1 we euce that µ(a) ν ( 1 ν(aɛ ) + ɛ ) ( ν(aɛ ) (1 + c 1 ɛ) + ɛ ) ν(a ɛ) + c 2 ɛ. µ 1 µ 1 ν 1 ν 1 ν 1 ν 1 ν 1 By symmetry, we fin For a function f : S 1 R we set P ( µ µ 1, ν ν 1 ) c 2 ɛ. f L := sup x y f(x) f(y), (29) x y f := sup f(x), f BL := f L + f. (30) x It follows from the proof of Corollary in [8] that ( µ f ν ) ( µ 2 f BL P, ν ). µ 1 ν 1 µ 1 ν 1 Thus, for any function f : S 1 R with f BL 1 we get [ ( f(µ ν) µ µ 1 f ν ) + 1 µ 1 ν 1 1 ν 1 µ 1 ] f ν µ 1 (2c 2 ɛ + c 3 ɛ) = c 4 ɛ. 12

13 We may assume that K RB, then h(k, ) BL 2R (cf. [23], Lemma ). Therefore, V (K) V 1 (L, K) = 1 h(k, u)(µ ν)(u) 2R S c 4ɛ = c 5 ɛ, 1 similarly V (L) V 1 (K, L) c 5 ɛ. These estimates correspon to the inequalities (7.2.6) in [23], an the proof can now be complete as the proof of Lemma an of Theorem in [23]. Note that the latter proof gives δ(k, L+x) cɛ 1/ if ɛ is smaller than a certain positive constant ɛ 1 epening only on, r, R; if ɛ ɛ 1, then the same inequality is achieve by a suitable choice of c. 4 Stability results in the Brunn-Minkowski-Firey theory A basic notion of the Brunn-Minkowski theory is the vector aition of convex boies, which correspons to the aition of support functions, h(k + L, ) = h(k, ) + h(l, ). For p 1, a p-sum of convex boies K, L K 0 (the set of convex boies in R with 0 as interior point) can be efine by h(k + p L, ) := [h(k, ) p + h(l, ) p ] 1/p, since the right-han sie is again a support function. Such p-means of convex boies were introuce by Firey [9]. Lutwak [19], [20] has extene large parts of the Brunn-Minkowski theory to this more general combination of convex boies. In this Brunn-Minkowski-Firey theory, as it is now calle, the role of the classical surface area measures S m (K, ), m = 0,..., 1 (see, e.g., Section 4.2 of [23]) is playe by measures S p,i (K, ) on the sphere S 1 (i = 0,..., 1). The measure S p,i (K, ) is absolutely continuous with respect to S 1 i (K, ) an has a Raon- Nikoym erivative given by S p,i (K, ) S 1 i (K, ) = h(k, )1 p. (Note that in [19] the measure S 1,i (K, ) is, unfortunately, enote by S i (K, ) an not by S 1 i (K, ), as usual.) Lutwak s theory contains an analogue of the Aleksanrov-Fenchel-Jessen theorem, Corollary (2.3) of [19]: Suppose K, L K 0 an 0 i <. If i p > 1 an S p,i (K, ) = S p,i (L, ), then K = L. In the following, we obtain a stability version of this result. Again, we use an assumption on the Prohorov istance of two measures, which is weaker than the corresponing assumption for the total variation istance of the measures. 13

14 By K 0(r, R) we enote the set of convex boies K R which satisfy rb K RB, where 0 < r < R. 4.1 Theorem. Let p > 1 an 0 < r < R. Suppose that K, L K 0(r, R), i {0,..., 1}, i p, an with some ɛ 0. Then P (S p,i (K, ), S p,i (L, )) ɛ (31) δ(k, L) cɛ q/2 with q = where the constant c epens only on, p, r, R. 1 ( + 1)2 i 2, Proof. In the following, c 1, c 2,... enote positive constants which epen only on, p, r, R. In the subsequent estimations where such constants occur, we very often tacitly use the facts that rb K, L RB, an that mixe volumes are monotone in each argument. With K an L as in the theorem, we use the notations (all integrations are over the sphere S 1 ) W i (K) = 1 h(k, u)s 1 i (K, u), W i (K, L) = 1 h(l, u)s 1 i (K, u) = V (K[ 1 i], L[1], B [i]), W p,i (K, L) = 1 h(l, u) p S p,i (K, u) = 1 h(l, u) p h(k, u) 1 p S 1 i (K, u). As in [23], p. 398, we write, for some fixe i {0,..., 1} an for k {0,..., i}, V (k) := V (K[ i k], L[k], B [i]), thus W i (K) = V (0), W i (K, L) = V (1), W i (L) = V ( i). With these notations, Lutwak s [19] inequality (IIp) (p. 132; see also Theorem 1.2 in [19]) reas Interchanging K an L, we get W p,i (K, L) i V i p (0) V p ( i). (32) W p,i (L, K) i V i p ( i) V p (0). (33) 14

15 Another inequality prove by Lutwak [19] (p. 137) states that W p,i (K, L) W i (K, L) p W i (K) 1 p = V p (1) V 1 p (0). (34) Using (31), we can estimate as in Section 3 an obtain V(0) W p,i (L, K) = 1 h(k, u) p [S p,i (K, u) S p,i (L, u)] c 1 h(k, ) p BL ɛ, hence an similarly We write V (0) W p,i (L, K) c 2 ɛ (35) W p,i (K, L) V ( i) c 3 ɛ. (36) W p,i (K, L) V i p i = ( V( i) + V (0) ( V( i) V (0) (0) V p i ( i) ) p i ( V i p i ( i) ) V p i (0) W p,i (L, K) ) p i [Wp,i (L, K) V (0) ] + [ Wp,i (K, L) V ( i) ]. By (33), the first term on the right is not positive, hence (35) an (36) give W p,i (K, L) V i p i (0) V p i ( i) c 4ɛ. (37) Now we assume that i {0,..., 2}. We write (37) in the form [ ] [ ] W p,i (K, L) V p (1) V 1 p (0) + V p p( 1 i) (1) V i (0) V p i ( i) V 1 p (0) c 4 ɛ. Here both brackets are nonnegative, the first by (34), an the secon by the Aleksanrov-Fenchel inequalities. We euce that an W p,i (K, L) V p (1) V 1 p (0) c 4 ɛ (38) V p p( 1 i) (1) V i Interchanging K an L in (39) gives V p (0) V p i ( i) + c 5ɛ. (39) p( 1 i) ( 1 i) V i ( i) V p i (0) + c 5 ɛ. (40) 15

16 Multiplication of (39) an (40) yiels V (1) V ( 1 i) V (0) V ( i) + c 6 ɛ. (41) We are now in the same situation as in the proof of Theorem in [23]: the inequality there before (7.2.12) is precisely (41), with m replace by i an c 2 replace by c 6. Hence, the subsequent arguments in [23] (see the Appenix of the present paper) lea to the conclusion that δ(k, L) c 7 ɛ q (42) (see also the hint at the en of the proof of Theorem 3.1). Here K = [K s(k)]/b(k), where s(k) is the Steiner point an b(k) is the mean with of K. We put λ = b(k)/b(l) an t = s(k) λs(l), then (42) implies δ(k, λl + t) c 8 ɛ q. (43) To erive (34), Höler s inequality was use. In orer to estimate t, we nee a sharper version of that inequality. We use a special case of an inequality by Kober [18], namely w 1 a 1 + w 2 a 2 a w1 1 aw2 2 w(a 1/2 1 a 1/2 2 ) 2 for a 1, a 2 0 an w 1, w 2 > 0 with w 1 + w 2 = 1, where w := min{w 1, w 2 }. Here we put, for p > 1 an a, b > 0, an obtain w 1 = 1 p, w 2 = p 1 p, a 1 = a p b 1 p, a 2 = b a p b 1 p + (p 1)b pa mb 1 p (a p/2 b p/2 ) 2 (44) with m = min{1, p 1}. Write h(m, ) = h M for M K an put I(M) := 1 h M (u) S 1 i (K, u). We apply (44) with a = h L(u) I(L), b = h K(u) I(K), where u S 1, an integrate over all u S 1 with respect to the measure (1/)S 1 i (K, ). The result can be written as W p,i (K, L) V p (1) V 1 p (0) [ ( ) p/2 ( ) ] p/2 2 hl hk 1 c 9 h 1 p K I(L) I(K) S 1 i(k, ). (45) 16

17 The quotient h L /I(L) is invariant uner a ilatation of L, hence on the right-han sie, the boy L can be replace by λl. Therefore, (38) an (45) yiel [ ( ) p/2 ( ) ] p/2 2 hλl hk S 1 i(k, ) c 10ɛ. (46) I(λL) I(K) Since I(M) is invariant uner translations of M, inequality (43) shows that I(λL) I(K) c 11 ɛ q. Using this inequality an the mean value theorem, we can estimate, for u S 1, h λl (u) h K (u) c 12 hλl (u) p/2 h K (u) p/2 ( ) p/2 hλl (u) c 13 I(λL) ( ) p/2 hk (u) I(K) + c 14ɛ q. Together with (46), this yiels an estimate ( c 15 ) 2 h λl h K S 1 i (K, ) h λl h K 2 S 1 i (K, ) c 16 ɛ q. Since h λl+t (u) = h λl (u) + u, t, it follows from (43) that u, t h λl (u) h K (u) + c 8 ɛ q for u S 1. Writing t 1 = t/ t if t 0, we euce that t u, t 1 S 1 i (K, u) c 17 ɛ q/2. Now u, t 1 S 1 i (K, u) c 18, since the integral is, up to a factor epening only on, an intrinsic volume of a projection of K an hence can be estimate from below by a constant epening only on an r. The conclusion is that t c 19 ɛ q/2. (47) To estimate λ, we first euce from (36) an (37) that V ( i) V i p i (0) V p i 17 ( i) c 20ɛ,

18 thus V i p i ( i) V i p i (0) c 21 ɛ. Since we have assume i p 0, this implies From (43), we get hence W i (L) W i (K) c 22 ɛ. K λl + t + c 8 ɛ q B (1 + c 23 ɛ q )λl + t, W i (L) W i (K) + c 22 ɛ W i ((1 + c 23 ɛ q )λl) + c 22 ɛ = [(1 + c 23 ɛ q )λ] i W i (L) + c 22 ɛ. This gives λ 1 c 24 ɛ q. Interchanging the roles of K an L, we similarly obtain λ 1 1 c 25 ɛ q an hence λ 1 c 26 ɛ q. (48) The inequalities (43), (47) an (48) finally give δ(k, L) c 27 ɛ q/2. This completes the proof of Theorem 4.1 in the case where i {0,..., 2}. Finally, we consier the (simpler) case i = 1. As before, we euce that W 1 (L) W 1 (K) c 22 ɛ, an hence by symmetry where λ = b(k)/b(l). Using (37) an (45), we fin that λ 1 c 28 ɛ, (49) [ ( ) p/2 ( ) ] p/2 2 h L h K σ c 29ɛ W 1 (L) W 1 (K) an thus h p/2 λl hp/2 K 2 σ c 30 ɛ. An application of the mean value theorem shows that h λl h K 2 σ c 31 ɛ, hence (49) an Corollary 1 in [25] (see also Lemma in [23]) give δ(k, L) c 32 ɛ The proof of Theorem 4.1 is now complete. 18

19 We remark that the preceing proof also permits us to give stability versions of two inequalities ue to Lutwak [19]. The first of these is his inequality (II p ) (which is (32) above). 4.2 Corollary. Let p > 1 an 0 < r < R. Suppose that K, L K 0(r, R), i {0,..., 1} an W p,i (K, L) W i (K) i p i W i (L) p i ɛ (50) with some ɛ 0. Then there is a constant c epening only on, p, r, R such that δ(k, λl) cɛ q/2, where λ = b(k)/b(l) an q is as in Theorem 4.1. Proof. Assume that i {0,..., 2}. Then the assumption (50) implies that δ(k, L) c 33 ɛ q, by the argument after equation (37). The subsequent argument in the proof of Theorem 4.1, which shows that s(k) λs(l) c 34 ɛ q/2, remains the same, hence δ(k, λl) c 35 ɛ q/2, as state. The case i = 1 can be treate as in the proof of Theorem 4.1. The next result gives a stability version of Lutwak s Corollary (1.3) (using his notations). 4.3 Corollary. Let p > 1, 0 < r < R an ϑ (0, 1). Suppose that K, L K 0(r, R), i {0,..., 1} an W i ((1 ϑ) K + p ϑ L) p i (1 ϑ)wi (K) p i ϑwi (L) p i ɛ (51) with some ɛ 0. Then there is a constant c epening only on, p, r, R such that δ(k, τl) c min{ϑ, 1 ϑ} q/2 ɛ q/2, where τ is a suitable positive constant an q is as in Theorem 4.1. Proof. Put M := (1 ϑ) K + p ϑ L. From the efinitions of p-sums an of the functionals W p,i, we obtain W i (M) = W p,i (M, (1 ϑ) K + p ϑ L) = (1 ϑ)w p,i (M, K) + ϑw p,i (M, L). Since M K0(r, R), we can apply Corollary 4.2 an euce that, with suitable numbers τ 1, τ 2 > 0, ] W i (M) (1 ϑ) [c 36 δ(m, τ 1 K) 2 q + Wi (M) i p i W i (K) p i +ϑ ] [c 37 δ(m, τ 2 L) 2 q + Wi (M) i p i W i (L) p i. 19

20 From this we infer that W i (M) p i (1 ϑ)wi (K) p i ϑwi (L) p i (1 ϑ)c 38 δ(m, τ 1 K) 2 q + ϑc39 δ(m, τ 2 L) 2 q min{ϑ, 1 ϑ}c 40 [δ(m, τ 1 K) + δ(m, τ 2 L)] 2 q min{ϑ, 1 ϑ}c 41 δ(τ 1 K, τ 2 L) 2 q min{ϑ, 1 ϑ}c42 δ(k, τl) 2 q. 5 Stability of inverse integral transforms The starting point of this section is formula (6), V 1 (K u ) = 1 u, v S 1 (K, v), u S 1, 2 S 1 which expresses the projection function u V 1 (K u ) of a convex boy K as the cosine transform of its area measure of orer 1. The stability result of Bourgain & Linenstrauss [2] is a quantitative version of the fact that two -imensional convex boies with the same centre of symmetry must be close if their projection functions are close. Groemer s [14] book contains a etaile presentation of this theorem an its proof (Theorem 5.5.7). In the present section, we use the metho of Bourgain an Linenstrauss to obtain stability estimates for the inversion of further integral transforms of area measures occurring in the geometry of convex boies. These integral transforms are of the following type. Let Φ : [ 1, 1] R be a boune, Borel measurable function. For a finite signe Borel measure µ on S 1, let (T Φ µ)(u) := Φ( u, v )µ(v) for u S 1. (52) S 1 For a boune measurable function f on S 1, the transform T Φ f is efine as T Φ µ for the signe measure µ = fσ, where σ enotes spherical Lebesgue measure. We nee a few facts about spherical harmonics, which can all be foun in [14]. If Y n is a spherical harmonic of egree n on S 1, then with T Φ Y n = a,n (Φ)Y n 1 a,n (Φ) = ω 1 Φ(t)Pn(t)(1 t 2 ) ( 3)/2 t, 1 where P n is the Legenre polynomial of imension an egree n (e.g., [14], Th ). Here ω k = kκ k is the area of the k-imensional unit ball, an κ k is its volume. The numbers a,n (Φ) are calle the multipliers of T Φ. 20

21 For f, g L 2 (S 1 ), the space of square integrable real functions on S 1, a scalar prouct is efine by (f, g) := fg σ, S 1 an the L 2 -norm by f := (f, f). Let {Y nj : j = 1,..., N(, n)} be an orthonormal basis of the real vector space of spherical harmonics of egree n N 0. For f L 2 (S 1 ), the relation f Y n (53) means that Y n = N(,n) j=1 n=0 (f, Y nj )Y nj, an the series in (53) is calle the conense harmonic expansion of f ([14], p. 72). Similarly, for a finite signe measure µ on S 1 we write µ Y n (54) if If (54) hols, then Y n = N(,n) j=1 T Φ µ n=0 ( ) Y nj µ Y nj. S 1 a,n (Φ)Y n. (55) n=0 The following theorem is only a slight extension of the result of Bourgain an Linenstrauss, to general transformations T Φ. For the reaer s convenience, we repeat the essential steps of the proof, in a simplifie form, to inicate where changes are necessary. Recall that the norm BL was efine by (30) an that µ T V enotes the total variation norm of the signe measure µ. 5.1 Theorem. Assume that the multipliers of the transformation T Φ satisfy a,0 (Φ) 0, a,n (Φ) 1 bn β for n N (56) with suitable b, β > 0. Let µ be a finite signe measure on S 1, an let F : S 1 R be a Lipschitz function. Then for each α (0, 1/(1 + β)) there is a constant c epening only on, Φ, α such that F µ c F BL µ 1 α T V T Φµ α. S 1 21

22 If µ is even an (56) hols for even n, then the same conclusion can be rawn. Proof. We choose b, β (epening on Φ) so that (56) hols. The constants c 1, c 2,... in the following epen only on, Φ, b, β, α an hence only on, Φ, α. It was the iea of Bourgain an Linenstrauss [2] to use the Poisson transform µ τ := 1 1 τ 2 ω S (1 + τ 1 2 2τ u, v ) µ(v), u /2 S 1, for 0 < τ < 1. We have (all integrations are over S 1 ) F τ µ = F µ τ σ an F µ (F F τ ) µ + F τ µ F F τ µ T V + F µ τ σ. (57) For τ 1/4, F F τ 2 +1 ω 1 2 F L (1 τ) log (58) ω 1 τ ([14], Lemma 5.5.8). Moreover, F µ τ σ F µ τ. (59) If (54) is the conense harmonic expansion of µ, then µ τ τ n Y n. n=0 The maximal value of the function g(x) = x β τ x for x > 0 is ( β/e log τ) β, hence n β τ n (1 τ) β for n N. Therefore, (56) gives ( ) β ( ) β β 1 τ e log τ τ n c 1 (1 τ) β a,n (Φ). Together with Parseval s relation, this yiels µ τ 2 = τ 2n Y n 2 c 2 1(1 τ) 2β n=0 22 n=0 ( ) β β e a,n (Φ) 2 Y n 2.

23 Now (55) shows that µ τ c 1 (1 τ) β T Φ µ. (60) From (57), (58), (59), (60) we get F µ c 2 F T Φ µ (1 τ) β 2 + c 3 F L µ T V (1 τ) log 1 τ ] c 4 F BL [ T Φ µ (1 τ) β 2 + µ T V (1 τ) log. 1 τ Since Φ is boune, we have T Φ µ c 5 µ T V. Therefore, we can fin a constant c 6 an a number τ [ 1 4, 1) such that For this τ an for α (0, 1) we get T Φ µ (1 τ) β = c 6 µ T V (1 τ) log 2 1 τ. ) 1 α F µ c 7 F BL (1 τ) (log 1 α(1+β) 2 µ 1 α T V 1 τ T Φµ α. If now α < 1/(1 + β), then we get F µ c 8 F BL µ 1 α T V T Φµ α. If the signe measure µ is even, then the components in (54) satisfy Y n = 0 for o n. Therefore, one can conclue as above. This completes the proof of Theorem 5.1. The geometric applications are of the following type. 5.2 Theorem. Let Φ an β be as in Theorem 5.1, let 0 < r < R. For γ (0, 1/(1 + β)), there is a constant c epening only on, Φ, γ, r, R with the following property. If K, L K (r, R) an then µ := S 1 (K, ) S 1 (L, ), (61) δ(k, L + x) c T Φ µ γ (62) with a suitable vector x R. If K an L are centrally symmetric an (56) hols for even n, then the same conclusion can be rawn. 23

24 Proof. The constants c 1, c 2,... in this proof epen only on, Φ, γ, r, R. We apply Theorem 5.1 with α = γ to the measure µ given by (61) an to F = h K, the support function of K. Without loss of generality, we assume that K RB. Since µ T V = S 1 (K, S 1 ) + S 1 (L, S 1 ) can be estimate from above by a constant epening only on R an an the same is true for h K BL (cf. [23], Lemma ), we get F µ c 1 T Φ µ α. By the geometric meaning of F an µ, this reas an interchanging K an L we get V (K) V 1 (L, K) c 2 T Φ µ α, V (L) V 1 (K, L) c 2 T Φ µ α. By a result of Diskant [4] (compare the remark at the en of the proof of Theorem 3.1), the two inequalities together imply V (K) V 1 (L, K) ɛ, V (L) V 1 (K, L) ɛ δ(k, L + x) c 3 ɛ 1/ for suitable x R, provie that ɛ ɛ 0, where ɛ 0 > 0 is a constant epening only on, r, R. If c 2 T Φ µ α ɛ 0, then we get δ(k, L + x) c 4 T Φ µ α/, an if c 2 T Φ µ α > ɛ 0, the same estimate hols if c 4 is chosen suitably. If K an L are centrally symmetric, then the signe measure µ is even. This completes the proof of Theorem 5.2. The special case of Theorems 5.1 an 5.2 treate by Bourgain an Linenstrauss concerne the cosine transform, where Φ(t) = 1 2 t for t [ 1, 1]. In that case, (56) hols for even n with β = (+2)/2. Hence, for convex boies K, L K (r, R) with the same centre of symmetry an for the ( 1)st projection function V 1 (K, u) = V 1 (K u ), u S 1, one gets δ(k, L) c V 1 (K, ) V 1 (L, ) γ (63) for γ (0, 2/( + 4)). It is natural to ask for similar results for the ith projection function, V i (K, u) = V i (K u ) = 1 u, v S i (K, v), u S 1. 2 S 1 24

25 By a well-known integral geometric formula, the convex boies K, L satisfy V i (K, ) = V i (L, ) if the projections of K an L on an i-imensional subspace always have the same i-imensional volume. For i = 1, a strong stability result was prove by Gooey an Groemer [13]. In two books, the question for corresponing generalizations was pose. Groemer [14], p. 222, writes that at present such stability estimates exist only in the cases i = 1 an i = 1. Garner [10] asks in his Problem 4.7 (p. 157) whether a stability result of the type (63) can be obtaine for 1 < i < 1. Curiously, a positive answer on the basis of publishe results coul have been given at the time when those books were written. In fact, the analytic part of the Bourgain-Linenstrauss [2] proof (just replace µ by µ i := S i (K, ) S i (L, ) in the first part of the proof of Theorem 5.2) gives V (0) V (i) c 2 T Φ µ i α, V (i+1) V (1) c 2 T Φ µ i α for α (0, 2/( + 4)) (if Φ(t) = 1 2 t ), where V (k) := V (K[i + 1 k], L[k], B [ 1 i]). As shown in [23] (Proof of Lemma 7.2.3), the inequalities V (0) V (i) ɛ, V (i+1) V (1) ɛ for K, L K (r, R) an some ɛ > 0 imply 0 V (1) V i/(i+1) (0) V 1/(i+1) (i+1) ( ) R r + 1 ɛ. One can now essentially use the proof of a stability result for the Aleksanrov- Fenchel-Jessen theorem ([23], Theorem 7.2.6) to euce the following. 5.3 Theorem. Let i {2,..., 2} an 0 < r < R, let K, L K (r, R) be convex boies which are centrally symmetric with the same centre. For ( ) 1 γ 0, ( + 1)( + 4)2 i 2, there exists a constant c epening only on, γ, r, R such that δ(k, L) c V i (K, ) V i (L, ) γ. We turn to other integral transforms of type T Φ which have occurre in geometric contexts. The sine transform is the transformation T Φ with Φ(t) = 1 t 2 for t [ 1, 1]. If K K is a convex boy an u S 1, then V ( 1) (K, u) := = 1 2( + 1) V 2 (K (u + tu)) t 25 S 1 1 u, v 2 S 1 (K, v) (64)

26 (see [21], p. 60); here 2V 2 (K ) is the ( 2)-imensional surface area of a ( 1)- imensional convex boy K. Thus the functional V ( 1) (K, ), the integrate surface area of parallel hyperplane sections, is, up to a factor, the sine transform of the surface area measure of K. The sine transform S is connecte with the cosine transform C an the spherical Raon transform R by the relation RC = κ 2 S, which is easily obtaine by a irect calculation. (For convex boies this implies, in view of (64), that Raon transforms of projection functions are connecte with sections; this interplay was stuie in greater generality by Gooey [11].) This relation implies corresponing relations for the multipliers: if then f Y n, n=0 Cf ζ,n Y n, n=0 κ 2 Sf Rf ρ,n ζ,n Y n. n=0 ρ,n Y n, For even n, we have ζ 1,n = O ( n (+2)/2), as remarke above, an ρ 1,n = O ( n ( 2)/2) (as follows from [14], Lemma an (3.4.19)), hence ρ 1,n ζ 1,n = O(n ). Thus, for the function Φ(t) = 1 t 2, assumption (56) hols for even n with β =. This gives the following result. 5.4 Theorem. Let 0 < r < R, let K, L K (r, R) be convex boies with the same centre of symmetry. For γ (0, 1/( + 1)), there exists a constant c epening only on, γ, r, R such that n=0 δ(k, L) c V ( 1) (K, ) V ( 1) (L, ) γ. The results involving the cosine or sine transform are necessarily restricte to centrally symmetric convex boies, since a transform T Φ µ with an even function Φ oes not contain information on the o part of the signe measure µ. We turn now to stability versions of some uniqueness theorems for not necessarily symmetric convex boies. Anikonov an Stepanov [1] have propose to consier, for K K an u S 1, besies the projection volume V 1 (K, u) = V 1 (K u ), also the area S(K, u) of the illuminate portion of K in irection u, that is, S(K, u) := S 1 (K, {v S 1 : u, v 0}). They showe that the combine functional F (K, u) := pv 1 (K, u) + qs(k, u), u S 1, with constants p, q (p, q, 2pκ 1 +qω 0) etermines the convex boy K uniquely up to a translation. They also prove a corresponing stability result in R 3. This, 26

27 however, is rather weak, since it assumes that the ifference F (K, ) F (L, ) is small in a norm that involves erivatives up to orer six. A stronger result can be obtaine with the ai of Theorem 5.2. In fact, we have F (K, ) = T Φ S 1 (K, ) with Φ = pφ 1 +qφ 2, where Φ 1 (t) = 1 2 t an Φ 2 = 1 [0,1]. Now, a,n (Φ 1 ) = 0 for o n, an a,n (Φ 1 ) 1 = O(n (+2)/2 ) for even n. On the other han, a,n (Φ 2 ) = 0 for even n > 0, an a,n (Φ 2 ) 1 = O(n /2 ) for o n (see [14], Lemma an (3.4.20)). It follows that Φ satisfies (56) with β = ( + 2)/2 (note that the assumption 2pκ 1 + qω 0 ensures that a,0 (Φ) 0). Hence, for any two convex boies K, L K (r, R), we have δ(k, L + x) c F (K, ) F (L, ) γ with a suitable vector x R, for γ (0, 2/( + 4)) an with c epening only on, p, q, γ, r, R. The last two transformations to be consiere stem from the part of theoretical stereology or geometric tomography where one is intereste in obtaining information on convex boies from lower imensional sections. The secon mean section boy M 2 (K) of a convex boy K K was introuce by Gooey an Weil [12]. It is efine by h(m 2 (K), ) = h(k E, ) µ 2 (E). E 2 Here E 2 is the affine Grassmannian of two-imensional planes in R an µ 2 is its motion invariant measure, normalize so that µ 2 ({E E 2 : E B }) = κ 2. Thus, M 2 (K) comprises information about the two-imensional sections of K, in integrate form. Gooey an Weil showe that two -imensional convex boies K an L with M 2 (K) = M 2 (L) iffer only by a translation, an they mentione briefly (on p. 429) that a corresponing stability version coul be obtaine. We will make this more explicit. For unit vectors u, v, let α(u, v) [0, π] enote the angle between u an v. Gooey an Weil (loc. cit., Corollary 2) prove that h(m 2 (K) t, u) = κ 2κ 2 ( ) α(u, v) sin α(u, v) S 1 ( K, v) (65) 2 κ S 1 for u S 1, where t is a suitable translation vector. Their proof (cf. Theorem 2) provies no information about this translation, but a relate remark of Gooey [11], p. 165, gives a hint. Denoting by z r+1 (K) the intrinsic (r + 1)st moment vector of K (see [23], p. 304), we have 1 h(m 2 (K) t, u) u σ(u) = z 1 (M 2 (K) t) = z 1 (M 2 (K)) t κ S 1 = κ 2κ 2 ( 2) κ z 1 (K) t. 27

28 But this must be the zero vector, as we see by using (65), Fubini s theorem, the fact that a vector integral of the form S 1 f( u, v ) u σ(u) is invariant uner rotations fixing v an hence is a multiple of v, an that v S S 1 1 ( K, v) = 0. We euce that the boy ( ) 2 κ M 2(K) := M 2 (K) z 1 (K), κ 2 κ 2 which we call the normalize secon mean section boy of K, satisfies h(m 2(K), u) = α(u, v) sin α(u, v) S 1 ( K, v) for u S 1. S 1 Thus, h(m 2(K), ) = T Φ S 1 ( K, ) with Φ(t) = (arccos t) 1 t 2 for t [ 1, 1]. From a computation in [12] (formula (4.10), together with the relation ( ) n + 3 c n (t) = P 3 n(t) between Gegenbauer an Legenre polynomials, see [14], p. 97) it follows that a,n (Φ) = ( c() (n 1)(n + 1) n!γ ( 1 2 (n + )) (n + 2)!Γ ( 1 2 (n + 2)) with a constant c() epening only on. From this we euce that (56) hols with β =. For the resulting stability estimate, we may now use the Hausorff istance also on the right-han sie: For γ (0, ( + 1)), there exists a constant c epening only on, r, R, γ such that, for K, L K (r, R), for a suitable vector x R. δ(k, L + x) cδ(m 2(K), M 2(L)) γ The origin of our last example is an investigation, [24], on the oriente mean normal measure of a stationary stochastic process of convex particles an its etermination from planar sections. There one has reason to consier the function efine by V ( 1) + (K, u) := H 2 ( u K (u + tu)) t for K K an u S 1 ; here H 2 enotes the ( 2)-imensional Hausorff measure an u K is the upper bounary of K in irection u, that is, the set of ) 2, 28

29 all bounary points of K at which there exists an outer unit normal vector v with u, v 0. By formula (17) of [24], V ( 1) + (K, ) = T Φ S 1 (K, ) with Φ(t) = 1 t 2 1 [0,1] (so that T Φ coul be calle the hemispherical sine transform). For even n N, the multipliers of T Φ are essentially those of the sine transform, namely a,n (Φ) = 1 2 a,n(ψ) for Ψ(t) = 1 t 2. Hence, as shown before Theorem 5.4, a,n (Φ) 1 = O(n ) for even n. For o n, the multipliers have not been etermine explicitly, but it has been shown that a,1 (Φ) 0 an, for o n 3, with 1 3 (n 2) f(n) a,n (Φ) = ω 1 ( 1)( + 1) ( + n 4) (n + 1)(n + 3) n + 3 < ( 1) (n 1)/2 f(n) n + 1 ([24], p. 36 together with (20) an (19)). From this, one obtains a,n (Φ) 1 = O(n /2 ) for o n. We euce that (56) hols with β =. Hence, for 0 < r < R an γ (0, 1/( + 1)) there exists a constant c epening only on, γ, r, R such that, for K K (r, R), with a suitable vector x R. 6 Appenix δ(k, L + x) c V ( 1) + (K, ) V ( 1) + (L, ) γ In the proofs of Theorems 4.1 an 5.3 we have referre to the proof of Theorem in [23], which in turn relies on inequality (6.4.9) of [23] (p. 335). The proof of (6.4.9) given there is not complete, as A. Giannopoulos has kinly pointe out. We take this opportunity to correct the error (using the same notations). The proof of (6.4.9) as given is correct if U 12 U 00 U 01 U 02 < 0; observe that Now, for λ 1, λ 2 0 also 0 V (λ 1 K 1 + λ 2 K 2, K 0, C) 2 U 2 01 U 00 U 11 0, U 2 02 U 00 U (66) V (λ 1 K 1 + λ 2 K 2, λ 1 K 1 + λ 2 K 2, C)V (K 0, K 0, C) = λ 2 1(U 2 01 U 00 U 11 ) + λ 2 2(U 2 02 U 00 U 22 ) 2λ 1 λ 2 (U 12 U 00 U 01 U 02 ). If U 12 U 00 U 01 U 02 > 0, we can euce (6.4.9) from this inequality. If U 12 U 00 U 01 U 02 = 0, (6.4.9) hols by (66). 29

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