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30
ANALYTIC SOLUTION TO THE BUSEMANNPETTY PROBLEM ON SECTIONS OF CONVEX BODIES
, 1999
"... We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyp ..."
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Cited by 61 (12 self)
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We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in Rn and leads to a unified analytic solution to the BusemannPetty problem: Suppose that K and L are two originsymmetric convex bodies in Rn such that the ((n − 1)dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (ndimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the BusemannPetty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n − 2)nd derivative of the parallel section functions. The affirmative answer to the BusemannPetty problem for n ≤ 4 and negative answer for n ≥ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.
Generalizations of the BusemannPetty problem for sections of convex bodies
 J. FUNCT. ANAL
, 2004
"... We present generalizations of the BusemannPetty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For 2 and 3dimensional sections, both negative and posi ..."
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Cited by 24 (8 self)
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We present generalizations of the BusemannPetty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For 2 and 3dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.
Generalized intersection bodies
 J. Funct. Anal
"... Abstract. We study the structures of two types of generalizations of intersectionbodies and the problem of whether they are in fact equivalent. Intersectionbodies were introduced by Lutwak and played a key role in the solution of the BusemannPetty problem. A natural geometric generalization of th ..."
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Cited by 17 (1 self)
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Abstract. We study the structures of two types of generalizations of intersectionbodies and the problem of whether they are in fact equivalent. Intersectionbodies were introduced by Lutwak and played a key role in the solution of the BusemannPetty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersectionbodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized BusemannPetty problem. We show that these classes share many identical structure properties, proving the same results using Integral Geometry techniques for Zhang’s class and Fourier transform techniques for Koldobsky’s class. Using a Functional Analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the Integral Geometry of the Grassmann Manifold. 1.
A positive solution to the BusemannPetty problem in R^4
, 1999
"... This paper presents the correct solution, namely, the BusemannPetty problem has a positive solution in R ..."
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Cited by 16 (2 self)
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This paper presents the correct solution, namely, the BusemannPetty problem has a positive solution in R
The BusemannPetty problem for arbitrary measures
"... Abstract. The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of genera ..."
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Cited by 15 (2 self)
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Abstract. The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the BusemannPetty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized BusemannPetty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in R n and solve a version of generalized BusemannPetty problem for sections by kdimensional subspaces. 1.
Valuations and BusemannPetty type problems
 Adv. Math
"... Projection and intersection bodies define continuous and GL(n) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann– Petty problem. We consider the question whether ΦK ⊆ ΦL impli ..."
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Cited by 12 (2 self)
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Projection and intersection bodies define continuous and GL(n) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann– Petty problem. We consider the question whether ΦK ⊆ ΦL implies V (K) ≤ V (L), where Φ is a homogeneous, continuous operator on convex or star bodies which is an SO(n) equivariant valuation. Important previous results for projection and intersection bodies are extended to a large class of valuations.
The BusemannPetty problem via spherical harmonics
 Advances of Math
"... Abstract. The BusemannPetty problem asks whether symmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller ndimensional volume. The solution has recently been completed, and the answer is affirmative if n ≤ 4 and negative if n ≥ 5. In this article we present ..."
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Cited by 9 (2 self)
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Abstract. The BusemannPetty problem asks whether symmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller ndimensional volume. The solution has recently been completed, and the answer is affirmative if n ≤ 4 and negative if n ≥ 5. In this article we present a short proof of the affirmative result and its generalization using the FunkHecke formula for spherical harmonics. 1.
THE COMPLEX BUSEMANNPETTY PROBLEM ON SECTIONS OF CONVEX BODIES
"... Abstract. The complex BusemannPetty problem asks whether origin symmetric convex bodies in C n with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4. 1. ..."
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Cited by 9 (2 self)
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Abstract. The complex BusemannPetty problem asks whether origin symmetric convex bodies in C n with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4. 1.
Sections of star bodies and the Fourier transform
 Proceedings of the AMSIMSSIAM Summer Research Conference in Harmonic Analysis, Mt Holyoke, 2001, Contemp. Math
, 2003
"... A new approach to the study of sections of star bodies, based on methods of Fourier analysis, has recently been developed. The idea is to express certain geometric properties of bodies in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. This ap ..."
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Cited by 5 (2 self)
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A new approach to the study of sections of star bodies, based on methods of Fourier analysis, has recently been developed. The idea is to express certain geometric properties of bodies in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. This approach has already led to several results including an analytic solution to the BusemannPetty problem on sections of convex bodies. In this article we bring these results together and present short proofs of major connections.