A new ellipsoid associated with convex bodies
type: Item
instance of: scholarly article
title: A new ellipsoid associated with convex bodies
zbMATH Open document ID: 0974.52008
DOI: 10.1215/S0012-7094-00-10432-2
author: Erwin Lutwak
author: Deane Yang
author: Gaoyong Zhang
published in: Duke Mathematical Journal
publication date: 2001-12-16T00:00:00Z
review text: The Legendre ellipsoid \(\Gamma_2K\) of a star body \(K\) in Euclidean space \(\mathbb{R}^n\) can be defined by its support function \[ h_{\Gamma_2 K}(u):= \left({n+2 \over V(K)}\int_K (u\cdot x)^2 dx\right)^{1\over 2} \] for \(u\in S^{n-1}\). The authors propose a new ellipsoid, \(\Gamma_{-2}K\), which can be defined by its radial function \[ \rho_{\Gamma_{-2} K}(u):= \left({1\over V(K)} \int_{S^{n-1}} (u\cdot v)^2 h_K^{-1} (v)dS_K(v) \right)^{-{1\over 2}} \] for \(u\in S^{n-1}\); here \(S_K\) is the classical surface area measure of \(K\). The motivation comes from Lutwak's dual Brunn-Minkowski theory: it is remarked that the Legendre ellipsoid \(\Gamma_2K\) properly belongs to that theory, while \(\Gamma_{-2}K\) is its dual counterpart in the classical Brunn-Minkowski theory. The authors present a neat parallel development of the known inequalities for the Legendre ellipsoid \(\Gamma_2K\) (with new proofs) and their newly established counterparts for the ellipsoid \(\Gamma_{-2}K\). Among the interesting series of volume inequalities for the new ellipsoid there is, surprisingly, even one for which the dual classical counterpart is still an open problem.
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1529063
zbMATH Keywords: dual Brunn-Minkowski theory
zbMATH Keywords: volume estimates
zbMATH Keywords: Legendre ellipsoid
MaRDI profile type: Publication
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full work available at URL: s0012-7094-00-10432-2
OpenAlex ID: W1999146895
description: scientific article; zbMATH DE number 1529063