Quermassintegrals of a random polytope in a convex body
type: Item
instance of: scholarly article
title: Quermassintegrals of a random polytope in a convex body
zbMATH Open document ID: 1030.52006
published in: Archiv der Mathematik
publication date: 2003-08-18T00:00:00Z
review text: In this interesting paper the authors extend results of Blaschke, Groemer, and Macbeath on the \(p\)th moment of the volume of a random polytope contained in a convex body. Their main result is the following: Let \(K\subset\mathbb{R}^{n}\) be a convex body with volume \(\left|K\right|=1\), and let \(C(x_{1},\dots ,x_{N})\) denote the convex hull of \(N\geq n+1\) points \(x_{1},\dots ,x_{N}\) independently and uniformly chosen in \(K\). Then, for every continuous and strictly increasing function \(f:\mathbb{R}^{+}\mathbb{\rightarrow R}^{+}\), and for \(0\leq i\leq n\), the mean value \[ E(K,N,f\circ W_{i})=\int_{K}\dots \int_{K}f\left[ W_{i}(C(x_{1},\dots ,x_{N}))\right] dx_{1}\dots dx_{N} \] is minimal among all convex bodies \(K\) of volume \(1\) when \(K\) is a ball (\(W_{i}\) being the \(i\)th quermassintegral of a compact convex set). Moreover, the ball is the unique minimizer when \(1\leq i\leq n\). The proof is based on Steiner symmetrization, and on a well-known characterization of the euclidean ball.
reviewed by: C. Peri
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1966466
zbMATH Keywords: quermassintegrals
zbMATH Keywords: random polytopes
author: Marianna Hartzoulaki
author: Grigoris Paouris
MaRDI profile type: Publication
full work available at URL: s00013-003-4593-4
OpenAlex ID: W122202530
DOI: 10.1007/S00013-003-4593-4
description: scientific article; zbMATH DE number 1966466