Cover and bounded parallel faces
type: Item
instance of: scholarly article
title: Cover and bounded parallel faces
zbMATH Open document ID: 0927.52004
DOI: 10.1023/A:1005064005857
author: Jacques Bair
author: J.-C.Dupin
published in: Geometriae Dedicata
publication date: 1999-12-06T00:00:00Z
review text: \textit{D. Gale} and \textit{V. Klee} [Math. Scand. 7, 379--391 (1960; Zbl 0115.16501)] showed that the conical hull of a ``continuous'' convex set from a point which does not belong to this convex set is closed. In 1962, \textit{C. C. Braunschweiger} and \textit{H. E. Clark} [Am. Math. Mon. 69, 272--277 (1962; Zbl 0116.08302)] asked the question: when is the conical hull \(\mathbb C(A,x)\) of a closed convex set \(A\) in \(\mathbb R^n\) from a point \(x\) closed? This problem was solved in 1983 by \textit{J. Bair} and \textit{F. Jongmans} [Bull. Soc. R. Sci. Liège 52, 285--294 (1983; Zbl 0526.52001)]: these authors proved the theorem of the closed conical hull by introducing the notion of cover. At first, the cover \(\mathbb G(A)\) of a closed convex set \(A\) was defined as the intersection of some ``support-cones'' of \(A\) from marginal rays and asymptotes. Now, several equivalent definitions are given for the cover, especially, the cover is the shadow from the recession cone. The authors use the cover or the closure of the cover in order to determine when there exists a translate \(H'\) (resp. \(H''\)) of the hyperplane \(H\) such that its intersection with a closed convex set \(A\) (resp. \(B\)) is a bounded face. It is shown that the study of the closure of the cover gives interesting results about the cover itself.
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1241003
zbMATH Keywords: convex set
zbMATH Keywords: cover
zbMATH Keywords: shadow
zbMATH Keywords: recession cone
zbMATH Keywords: conical hull
zbMATH Keywords: strong separation
zbMATH Keywords: strict separation
zbMATH Keywords: parallel faces
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1241003