Stability of the reverse Blaschke-Santaló inequality for zonoids and applications
type: Item
instance of: scholarly article
title: Stability of the reverse Blaschke-Santaló inequality for zonoids and applications
zbMATH Open document ID: 1191.52002
author: Károly jun. Böröczky
author: Daniel Hug
published in: Advances in Applied Mathematics
publication date: 2010-04-01T00:00:00Z
review text: Given a centered convex body \(A\subset {\mathbb R}^n\), the volume product \(P(A)\) is defined as follows: \(P(A) :=|A|\cdot |A^*|\); here \(|\cdot|\) denotes volume and \(A^*\) is the polar body of \(A\). If \(A\) is a centered zonoid, then it is known that \(P(A)\geq P(C^n)\), where \(C^n\) is the centered affine cube. The equality here holds in the class of centered zonoids if and only if \(A=C^n\). The Mahler conjecture for centered convex bodies states that \(P(A)\geq P(C^n)\). The main purpose of this paper is to establish a strengthening of Mahler's estimate in terms of a stability result for convex zonoids, and to apply this to a problem in stochastic geometry. From the result obtained it follows that if \(P(A)\) is close to the lower bound of \(A\), then \(A\) must be close to some centered affine cube. Namely, there is a constant \(\gamma_n=\gamma_n(n)\) such that the following is true. If \(A\subset {\mathbb R}^n\) is a centered zonoid satisfying \(P(A)\leq (1+\varepsilon)P(C^n)\) for some \(\varepsilon\in [0,1]\), then \(\delta_{\text{BM}}(A,C^n)\leq 1+\gamma_n \varepsilon ^{1/n}\); \(\delta_{\text{BM}}(\cdot,\cdot)\) being the Banach-Mazur distance. This is applied in stochastic geometry as follows: We consider the zero cell \(Z_0\) of a stationary Poisson hyperplane mosaic \(\widehat X\) in \({\mathbb R}^n\). The expected number of vertices of \(Z_0\) is known to be minimal if \(\widehat X\) is a parallel mozaic, and maximal if \(\widehat X\) is isotopic. The authors show in a quantitative form that \(\widehat X\) is close to a parallel mozaic if the expected number of vertices of \(Z_0\) is almost minimal. A similar result is obtained for the upper bound.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 5689299
zbMATH Keywords: volume product
zbMATH Keywords: reverse Blaschke-Santaló inequality
zbMATH Keywords: Mahler's conjecture
zbMATH Keywords: zonoid
zbMATH Keywords: stability result
zbMATH Keywords: Poisson hyperplane tessellation
zbMATH Keywords: zero cell
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: j.aam.2009.09.002
OpenAlex ID: W1976102239
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DOI: 10.1016/J.AAM.2009.09.002
description: scientific article; zbMATH DE number 5689299