New affine inequalities and projection mean ellipsoids
type: Item
instance of: scholarly article
title: New affine inequalities and projection mean ellipsoids
zbMATH Open document ID: 1414.52005
author: Du Zou
author: Ge Xiong
published in: Calculus of Variations and Partial Differential Equations
publication date: 2019-02-14T00:00:00Z
review text: The authors introduce the notion of the projection mean ellipsoid $P_jK$ of a convex body $K\subset\mathbb{R}^n$, $1\leq j\leq n-1$. This is the unique origin-symmetric ellipsoid $E$ which maximizes volume subject to a constraint $\bar{\Lambda}_j(K,E)\leq 1$, the latter being related to the $j$th projection function $V_j(K|\cdot)$ of $K$. With this notion, the authors establish an inequality for the affine quermassintegrals \[ \Lambda_j(K)=n\frac{\omega_n}{\omega_j}\left(\int_{G_{n,j}}V_j(K|\xi)^{-n}\mathrm{d}\mu_j(\xi)\right)^{-1/n}, \] with $\mu_j$ being the normalized Haar measure on the Grassmannian $G_{n,j}$, $j\in\{1,\ldots,n-1\}$. Namely, \[ \Lambda_j(K)^n\geq n^n\omega_n^{n-j}V_n(P_jK)^j \] for $K\subset\mathbb{R}^n$ a convex body and $j\in\{1,\ldots,n-1\}$. (For $j\in\{2,\ldots,n-1\}$, equality holds precisely for the ellipsoid, and for $j=1$, equality holds iff $K$ has an $SL(n)$ image of constant width.) Furthermore, the authors show that \[ V_n(K^\ast)V_n(P_1K)\leq\omega_n^2 \] for $K=-K\subset\mathbb{R}^n$ a convex body, with equality precisely for the ellipsoid.
reviewed by: Thomas Jahn
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 46B06
zbMATH DE Number: 7023785
zbMATH Keywords: affine quermassintegral
zbMATH Keywords: ellipsoid
zbMATH Keywords: projection
zbMATH Keywords: Lutwak affine surface area
zbMATH Keywords: sharp isoperimetric inequality
MaRDI profile type: Publication
full work available at URL: s00526-019-1497-0
OpenAlex ID: W2912211307
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Wikidata QID: Q128489938
DOI: 10.1007/S00526-019-1497-0
description: scientific article; zbMATH DE number 7023785