Decomposition of closed convex sets
type: Item
instance of: scholarly article
title: Decomposition of closed convex sets
zbMATH Open document ID: 0661.52003
DOI: 10.1007/BF00147453
author: John André Wieacker
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: \textit{W. Weil} [Arch. Math. 34, 283-288 (1980; Zbl 0424.52002)] proved that a convex body L is a summand of a convex body K iff for each translate L'\(\not\subset K\) of L, the set of all points of L' farthest from K is a support set of L'. The author generalizes this result to the family \(C^ d\) of (not necessarily bounded) non-empty convex sets in \({\mathbb{R}}^ d\). He defines \(L\in C^ d\) to be a pseudo-summand of \(K\in C^ d\) if K is the closure of \(L+M\) for some \(M\in C^ d\). Then the following theorem is proved: \(L\in C^ d\) is a pseudosummand of \(K\in C^ d\) iff for each translate L' of L there is exactly one point in D(L',K) farthest from the origin. Here D(L',K) denotes the closure of the set of vectors \(x-p_ K(x)\), \(x\in L\), where \(p_ K\) denotes the nearest-point-map with respect to K.
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 4080284
zbMATH Keywords: summand of a convex set
reviewed by: Bernd Kind
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147453
OpenAlex ID: W1969348755
description: scientific article; zbMATH DE number 4080284