Minimal and reduced pairs of convex bodies
type: Item
instance of: scholarly article
title: Minimal and reduced pairs of convex bodies
zbMATH Open document ID: 0864.52002
DOI: 10.1007/BF00147810
author: Christina Bauer
published in: Geometriae Dedicata
publication date: 1997-06-24T00:00:00Z
review text: Call two pairs of convex bodies \((K,M)\) and \((K',M')\) in \(\mathbb{R}^d\) equivalent if the Minkowski sums \(K'+M\) and \(K+M'\) are equal. A pair \((K,M)\) is minimal if there is no equivalent pair \((K',M')\) with \(K'\varsubsetneq K\), \(M'\subseteq M\) or \(K'\subseteq K\), \(M'\varsubsetneq M\). A pair \((K,M)\) is called reduced if, whenever \(K'+M=K+M'=C\), there is a body \(L\) such that \(K+M+L=C\); reduced pairs are minimal. The author gives a new proof, using surface area measures, (and avoiding the use of Zorn's lemma) that equivalence classes of pairs of convex bodies in the plane always contain minimal elements, and a new proof of uniqueness of minimal pairs (up to translation). New characterizations of minimal and reduced planar pairs are also given, in terms of surface area measures. As a corollary of the first of these, it is shown that a pair of convex polygons is minimal iff then have at most one pair of equiparallel edges. In the final section, this corollary is generalized to pairs of polytopes in higher-dimensional spaces. Conditions for minimality are given, and it is shown that the reduced pairs of polytopes are precisely those with no equiparallel edges. Using the density of polytopes, it is shown that most pairs of convex bodies in \(\mathbb{R}^d\) are reduced, and the unique (up to translation) minimal members of their equivalence class.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 929898
zbMATH Keywords: Hörmander-Rädström lattice
zbMATH Keywords: minimal pairs
zbMATH Keywords: reduced pairs
zbMATH Keywords: surface area measure
zbMATH Keywords: convex bodies
reviewed by: Robert Dawson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00147810
OpenAlex ID: W2056681637
description: scientific article; zbMATH DE number 929898