The dual \(\phi\)-Brunn-Minkowski inequality
type: Item
instance of: scholarly article
title: The dual \(\phi\)-Brunn-Minkowski inequality
zbMATH Open document ID: 1479.52009
author: Wei Shi
author: Tian Li
author: Weidong Wang
published in: Mediterranean Journal of Mathematics
publication date: 2021-09-20T00:00:00Z
review text: Let \(K, L\subset\mathbb{R}^n\) be star bodies (i.e., star-shaped sets with respect to the origin, having continuous radial function), and let \(\rho_K\) and \(\rho_L\) be the radial functions of \(K\) and \(L\), respectively. Let \(\phi\in C^1((0,\infty))\) be strictly decreasing and concave, and assume that \(\log\circ\phi^{-1}\) is a convex function. For \(\xi\in[0,1]\), the authors define the function \(\rho_\xi\colon\mathbb{S}^{n-1}\to\mathbb{R}\) as follows: \[ \rho_\xi(u)=\sup\left\{\tau>0\colon (1-\xi)\phi\left(\frac{\rho_K(u)}{\tau}\right)+\xi\phi\left(\frac{\rho_L(u)}{\tau}\right)\le\phi(1)\right\}. \] The authors denote by \(\tilde Q_{\phi,\xi}\) the star body having \(\rho_\xi\) as radial function. The following Brunn-Minkowski-type inequality is proved: \[ (1-\xi)\phi\left(\frac{ |K|^{1/n}}{|\tilde Q_{\phi,\xi}|^{1/n}}\right)+ \xi\phi\left(\frac{|L|^{1/n}}{|\tilde Q_{\phi,\xi}|^{1/n}}\right)\le\phi(1), \] here \(|\cdot|\) denotes the volume, i.e., the Lebesgue measure in \(\mathbb{R}^n\). Moreover, when \(\xi\in(0,1)\), equality holds if and only if \(K\) and \(L\) are dilates of each other. The second main result of the paper is that the previous inequality is equivalent to the following Minkowski-type inequality: \[ \int_{\mathbb{S}^{n-1}}\phi\left(\frac{\rho_L(u)}{\rho_K(u)}\right)\, d\tilde V_K(u)\ge \phi\left(\frac{|L|^{1/n}}{|K|^{1/n}}\right), \] where \(\tilde V_K\) denotes the so-called \textit{dual cone-volume probability measure} of \(K\).
reviewed by: A. Colesanti
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A39
zbMATH DE Number: 7397593
zbMATH Keywords: star body
zbMATH Keywords: dual combination
zbMATH Keywords: dual Minkowski mixed volume inequality
zbMATH Keywords: dual Brunn-Minkowski inequality
MaRDI profile type: Publication
full work available at URL: s00009-021-01834-1
OpenAlex ID: W3196828731
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DOI: 10.1007/S00009-021-01834-1
description: scientific article; zbMATH DE number 7397593