Maximum and minimum sets for some geometric mean values
type: Item
instance of: scholarly article
title: Maximum and minimum sets for some geometric mean values
zbMATH Open document ID: 0715.60010
DOI: 10.1007/BF01045156
author: Richard E. Pfiefer
published in: Journal of Theoretical Probability
publication date: 1990-01-01T00:00:00Z
review text: Let \(v_ r\) denote the r-dimensional volume of the convex hull of \(r+1\) random points chosen independently and uniformly from a compact set K in \({\mathbb{R}}^ n\), and let h be a strictly increasing function. The author shows that for \(r=n\) (for \(r<n)\) the mean value of \(h\circ v_ r\) is minimal among all compact sets K of volume one, if K differs from an ellipsoid (from a ball) by a set of measure zero. This result is well- known only for convex K. Some other inequalities from geometric probability are extended to arbitrary compact sets. To prove these results, some integral inequalities for centrally symmetric quasiconvex and quasiconcave functions over compact rectangles are derived.
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 4178209
zbMATH Keywords: random simplex
zbMATH Keywords: minimal volume
zbMATH Keywords: characterization of ellipsoids
zbMATH Keywords: geometric probability
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4178209