An inequality between the parts into which a convex body is divided by a plane section
type: Item
instance of: scholarly article
title: An inequality between the parts into which a convex body is divided by a plane section
zbMATH Open document ID: 0531.52005
DOI: 10.1007/BF02851108
author: L. A. Santaló
publication date: 1983-01-01T00:00:00Z
review text: \textit{J. Bokowski} and \textit{E. Sperner jun.} [J. Reine Angew. Math. 311/312, 80-100 (1979; Zbl 0415.52008) found an upper bound for the product of the volumes of the two parts into which a hyperplane \(L_ n- 1\) splits a convex body K in Euclidean n-space. The bound is a constant times \(D^{n+1}\sigma_ n-1\) where D is the diameter of K, \(\sigma_ n-1\) is the (n-1)-dimensional volume of \(K\cap L_ n-1\). The present paper provides a different, integral-geometric proof for the cases \(n=2, 3\) and this proof is generalized to provide the corresponding inequalities in the cases of elliptic and hyperbolic spaces of dimensions 2 and 3.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 3842626
zbMATH Keywords: product of volumes
zbMATH Keywords: diameter
reviewed by: William J. Firey
MaRDI profile type: Publication
cites work: Zerlegung konvexer Körper durch minimale Trennflächen.
cites work: Ungleichungen für den Inhalt von Trennflächen
cites work: Q4110292
full work available at URL: https://doi.org/10.1007/bf02851108
OpenAlex ID: W1990609905
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3842626