A generalization of the Busemann-Petty problem on sections of convex bodies
type: Item
instance of: scholarly article
title: A generalization of the Busemann-Petty problem on sections of convex bodies
zbMATH Open document ID: 0938.52009
DOI: 10.1007/BF02808176
published in: Israel Journal of Mathematics
publication date: 2000-06-21T00:00:00Z
review text: For a convex body \(K\) in Euclidean space \(\mathbb{R}^n\) and for a unit vector \(\xi\), the parallel section function \(A_{K,\xi}\) is defined by \[ A_{K,\xi}(z): =\text{vol}_{n-1} \bigl(K\cap (\xi^\perp+ z\xi)\bigr)\quad \text{for }z\in\mathbb{R}. \] The main result of this paper is the following theorem. Let \(K\) and \(L\) be \((k-1)\)-smooth origin symmetric convex bodies in \(\mathbb{R}^n\) such that, for every unit vector \(\xi\), \[ (-1)^{(k-1)/2}A_{k, \xi}^{(k-1)}(0)\leq(-1)^{(k-1)/2} A_{L,\xi}^{(k-1)}(0), \] where \(k\in\{1,\dots, n-1\}\) is an odd integer and \(f^{(k-1)}\) denotes the \((k-1)\)st derivative of \(f\). Then (i) if \(k\geq n-3\) then \(\text{vol}_n(K) \leq\text{vol}_n(L)\), (ii) if \(k<n-3\) then it is still possible that \(\text{vol}_n(K)> \text{vol}_n (L)\). This theorem is closely related to the (known) solution of the Busemann-Petty problem; in fact, it implies its affirmative answer for \(n=4\) and its negative answer for \(n=5\). The proof of the theorem uses Fourier analysis. A crucial fact in the solution of the Busemann-Petty problem was the connection between this problem and intersection bodies, due to Lutwak. The paper contains two further results generalizing this connection, as well as a generalization of intersection bodies.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 1310633
zbMATH Keywords: section volume
zbMATH Keywords: intersection body
zbMATH Keywords: Parseval's formula
zbMATH Keywords: Busemann-Petty problem
author: Alexander Koldobsky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1310633