On the volume product of polygons
type: Item
instance of: scholarly article
title: On the volume product of polygons
zbMATH Open document ID: 1232.52009
author: Mathieu Meyer
author: Shlomo Reisner
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2011-11-04T00:00:00Z
review text: The volume product of a convex body in Euclidean space is the product of its volume and the volume of its polar body where the origin is chosen so that this is minimal. The Blaschke-Santaló inequality says that the volume product attains its maximum for ellipsoids. Here it is shown that in the plane the volume product of convex polygons with at most \(n\) vertices is attained by the affine images of regular polygons with \(n\) vertices. The authors also give new proofs of Mahler's inequalities which assert that lower bounds are attained by triangles or, in the centrally symmetric case, by parallelograms. The corresponding problem in dimensions greater than 2 is still unsolved, see, e.g. [\textit{R. Schneider}, Convex bodies: the Brunn-Minkowski theory. Cambridge: Cambridge University Press (1993; Zbl 0798.52001); Convex bodies: the Brunn-Minkowski theory. Paperback reprint of the hardback edition 1993, Cambridge: Cambridge University Press (2008; Zbl 1143.52002)] or \textit{K. J. Böröczky} and \textit{D. Hug} [Adv. Appl. Math. 44, No.~4, 309--328 (2010; Zbl 1191.52002)].
reviewed by: Erhard Heil
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 5968110
zbMATH Keywords: convex bodies
zbMATH Keywords: volume-product
zbMATH Keywords: polygons
zbMATH Keywords: affinely-regular
zbMATH Keywords: Blaschke-Santaló inequality
zbMATH Keywords: Mahler inequality
MaRDI profile type: Publication
full work available at URL: s12188-011-0054-3
OpenAlex ID: W2078165971
cites work: Affinely regular polygons
cites work: Properties of affinely regular polygons
cites work: LOCAL MINIMALITY OF THE VOLUME‐PRODUCT AT THE SIMPLEX
cites work: Q5771556
cites work: A theorem on transfer for convex bodies
cites work: Convex bodies with minimal volume product in \(\mathbb{R}^ 2\)
cites work: On the Blaschke-Santaló inequality
DOI: 10.1007/S12188-011-0054-3
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description: scientific article; zbMATH DE number 5968110