Das isodiametrische Problem der Minkowski-Geometrie. (The isodiametric problem in Minkowskian geometry)
type: Item
instance of: scholarly article
title: Das isodiametrische Problem der Minkowski-Geometrie. (The isodiametric problem in Minkowskian geometry)
zbMATH Open document ID: 0633.52010
DOI: 10.1007/BF03322393
author: Woldemar Barthel
author: H. Pabel
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The paper deals with some inequalities in the style of the classical inequalities of Minkowski for convex bodies: a) If A and B are two non-void compact sets in affine space \(A^ n\) with standard body (Eichfigur) (o,E) the inequality \(| A|^{1/n}+ | B|^{1/n}\leq D(A,B;E)| E|^{1/n}\) holds, where \(| X|\) means the Lebesgue measure of X and D(A,B;E) is the maximal distance between A and B with the Minkowski metric induced by the standard body (o,E); b) For any non-void measurable set A in affine space \(A^ n\) with standard body (o,E), the inequality \(D(A,E)\geq 2| E|^{1/n}| A|^{1/n}\) holds. The cases of inequality are discussed and several consequences are pointed out. For \(n=3\), noteworthy applications to computer images related with crystallography are given.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 4030372
zbMATH Keywords: Minkowski metric
zbMATH Keywords: distance between sets
zbMATH Keywords: isodiametric problem
zbMATH Keywords: Lebesgue measure
reviewed by: L. A. Santaló
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4030372