The LYZ centroid conjecture for star bodies
type: Item
instance of: scholarly article
title: The LYZ centroid conjecture for star bodies
zbMATH Open document ID: 1404.52006
author: Denghui Wu
author: Jiazu Zhou
published in: Science China. Mathematics
publication date: 2018-08-10T00:00:00Z
review text: For the beginning, the authors give a nice review of centroid bodies theory, in particular they formulate the LYZ Conjecture: If \( \phi \in {\mathcal C}\) and \( K \) is a star body in \( {\mathbb R }^n \) that contains the origin in its interior, then the volume ratio \( |\Gamma_{\phi} K| / |K| \) is minimized if and only if \( K \) is an ellipsoid centered at the origin. Let \( {\mathcal C} \) denote the set of convex functions \( \phi : {\mathbb R} \to [0,\infty)\) such that \( \phi (0) = 0\) and \( \phi (t) + \phi (-t) \neq 0 \) for \( t \neq 0 \), \( S^n_{o}\) be the set of star bodies in \( {\mathbb R }^n \) with respect to the origin. Let \( \Gamma_{\phi} K \) be the Orlicz centroid body for \( \phi \in {\mathcal C} \), \( x \in {\mathbb R }^n \) defined by the support function \[ h_{\Gamma_{\phi} K } (x) = \inf \left\{ \lambda > 0 : \frac{1}{|K|} \int_K \phi \left( \frac{x \cdot y}{\lambda}\right) dy \leq 1 \right\}. \] The authors study the LYZ Conjecture ``for star bodies in \( \tilde{S}^n_{o} \) without the condition of \( \phi\)'s strict convexity''. The main result of the paper is: Theorem 1.1. Let \( \phi \in {\mathcal C}\) and \( K \) be a star body with respect to the origin. Then the volume ratio \( |\Gamma_{\phi} K| / |K| \) is minimized when \( K \) is an ellipsoid centered at the origin. If \( K \in \tilde{S}^n_{o}\), then ellipsoids centered at the origin are the only minimizers. The authors reduce the statement of this theorem to the case of convex body \( \phi \) considered by E. Lutwak.
reviewed by: Vladimir Golubyatnikov
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 6916566
zbMATH Keywords: Brunn-Minkowski theory
zbMATH Keywords: Orlicz centroid body
zbMATH Keywords: Orlicz centroid inequality
zbMATH Keywords: star body
zbMATH Keywords: Steiner symmetrization
Wikidata QID: Q123363272
MaRDI profile type: Publication
full work available at URL: s11425-016-9199-2
OpenAlex ID: W2799948097
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DOI: 10.1007/S11425-016-9199-2
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