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21
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.
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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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.
COMPARISON OF VOLUMES OF CONVEX BODIES IN REAL, COMPLEX, AND QUATERNIONIC SPACES
, 2009
"... The classical BusemannPetty problem (1956) asks, whether originsymmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ≤ 4 and negative if n> 4. The same question can be asked when volumes of hy ..."
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Cited by 5 (0 self)
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The classical BusemannPetty problem (1956) asks, whether originsymmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ≤ 4 and negative if n> 4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic ndimensional spaces. All cases are treated simultaneously. In particular, we show that the BusemannPetty problem in the quaternionic ndimensional space has an affirmative answer if and only if n = 2. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations are discussed.
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.
Gaussian Measure of Sections of Convex Bodies
 Adv. Math
"... In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the BusemannPetty problem fo ..."
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Cited by 5 (3 self)
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In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the BusemannPetty problem for Gaussian measures. Key words: Convex body, Gaussian Measure, BusemannPetty problem 1