The spherical conjecture in Minkowski geometry
type: Item
instance of: scholarly article
title: The spherical conjecture in Minkowski geometry
zbMATH Open document ID: 1060.52003
author: Achill Schürmann
published in: Archiv der Mathematik
publication date: 2005-01-20T00:00:00Z
review text: In 1991 Croft, Falconer and Guy wrote ``\dots packing \(m\) disks or balls into a convex body so that the perimeter or surface area, or alternatively the diameter is minimum, it seems that the optimum shape is roughly circular or spherical if \(m\) is large.'' This ``spherical conjecture'' was independently verified by Böröczky and Zong in 1994. Both prove that the conjecture is true in a more general setting: whenever a quermassintegral different from the volume is minimum. In this note the author shows that the ``spherical conjecture'' is also valid in Minkowski geometry. In particular he proves that the shapes of convex bodies containing \(m\) translates of a convex body \(K\) so that their Minkowskian surface area is minimum tend to a convex body \(L\) which depends on the chosen Minkowskian space.
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 51B20
Mathematics Subject Classification ID: 05B40
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 2130541
zbMATH Keywords: Minkowski geometry
zbMATH Keywords: finite packings
zbMATH Keywords: optimum shape
zbMATH Keywords: Minkowskian surface area
zbMATH Keywords: convex body
MaRDI profile type: Publication
full work available at URL: s00013-004-4617-0
OpenAlex ID: W36492929
Wikidata QID: Q122913585
DOI: 10.1007/S00013-004-4617-0
description: scientific article; zbMATH DE number 2130541