An asymmetric Orlicz centroid inequality for probability measures
type: Item
instance of: scholarly article
title: An asymmetric Orlicz centroid inequality for probability measures
zbMATH Open document ID: 1306.52003
author: Qingzhong Huang
author: Binwu He
published in: Science China. Mathematics
publication date: 2014-12-02T00:00:00Z
review text: Let \(\operatorname{Conv}(\mathbb{R})\) be the class of convex functions \(\phi: \mathbb{R}\to [0, \infty)\), \(\phi(0)=0\) and such that \(\phi\) is either strictly decreasing on \((-\infty, 0]\) or \(\phi\) is strictly increasing on \([0, \infty)\). Denote by \(D_n\) the Euclidean ball of volume one. Lebesgue measure restricted to \(D_n\) is \(\lambda_{D_n}\). Let \({\mathcal{P}}_{[n]}\) be the class of probability measures on \(\mathbb{R}^n\) that are absolutely continuous with respect to Lebesgue measure. For a convex body \(K\) (that is a compact convex set in \(\mathbb{R}^n\) with non-empty interior) denote by \(h(K, y) =\max\{x\cdot y: x\in K\}\), for \(y\in K\), the support function of \(K\). In [\textit{E. Lutwak} et al., J. Differ. Geom. 84, No. 2, 365--387 (2010; Zbl 1206.49050)] has been proved the following Orlicz centroid inequality for convex bodies: Let \(\phi\in \operatorname{Conv}(\mathbb{R})\) and \(K\) be a convex body in \(\mathbb{R}^n\) that contains the origin in its interior with \(vol_n(K)=1\). Define the Orlicz centroid body \(\Gamma_{\phi}K\) to be the convex body whose support function at \(y\in \mathbb{R}^n\) is given by \(h(\Gamma_{\phi}K, y) = \inf\left\{\lambda>0: \int_K\phi\left(x\cdot y/\lambda\right)\, dx\leq 1\right\}\). Then, \(vol_n(\Gamma_{\phi}K) \geq vol_n(\Gamma_{\phi}D_n)\), equality holds if and only if \(K\) is an ellipsoid centered at the origin. Haberl and Schuster's asymmetric \(L_p\) centroid inequality in [\textit{C. Haberl} and \textit{F. E. Schuster}, J. Differ. Geom. 83, No. 1, 1--26 (2009; Zbl 1185.52005)] is the case \(\phi(t) =(|t| +\alpha t)^p\), for \(-1\leq \alpha \leq 1\) of the inequality. By using probability arguments, \textit{G. Paouris} and \textit{P. Pivovarov} in [Adv. Math. 230, No. 3, 1402--1422 (2012; Zbl 1260.52006)] obtained the symmetric Orlicz centroid inequality for probability measures: Let \(\phi\in \operatorname{Conv}[0, \infty)\) and \(\mu\in {\mathcal{P}}_{[n]}\). Define the symmetric Or…
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 6376114
zbMATH Keywords: \(M\)-addition
zbMATH Keywords: Orlicz centroid inequality
zbMATH Keywords: asymmetric Orlicz centroid bodies
zbMATH Keywords: asymmetric \(L_p\) centroid bodies
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
full work available at URL: s11425-014-4785-3
OpenAlex ID: W1982765041
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DOI: 10.1007/S11425-014-4785-3
description: scientific article; zbMATH DE number 6376114