An extension of the isoperimetric inequality on the sphere
type: Item
instance of: scholarly article
title: An extension of the isoperimetric inequality on the sphere
zbMATH Open document ID: 0714.52003
author: Béla Bollobás
published in: Elemente der Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/141464
review text: For a subset A of the n-sphere \(S^ n\) in \({\mathbb{R}}^{n+1}\) the author defines the k-diameter of A to be the supremum of the \(\min \{d(x_ i,x_ j)|\) \(1\leq i<j\leq k\}\) where the supremum is taken over every set of k points \(x_ 1,...,x_ k\) of A. Here d(\(\cdot,\cdot)\) denotes the spherical distance on \(S^ n\). The 2-diameter of A is then the ordinary diameter of A. He shows that if \(C\subset S^ n\) is a spherical cap having the same Lebesgue measure as A then the k-diameter of A is not less than the k-diameter of the cap C. The proof follows Benyamini's [\textit{Y. Benyamini}, Longhorn Notes, Univ. Texas, Texas Funct. Anal. Semin., 53-76 (1983-84)] proof of the classical isoperimetric inequality on the sphere using compression operators.
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 4175882
zbMATH Keywords: diameter
zbMATH Keywords: isoperimetric inequality
reviewed by: Bernd Kind
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4175882