More on compacta with convex projections.
type: Item
instance of: scholarly article
title: More on compacta with convex projections.
zbMATH Open document ID: 1044.52005
author: Stoyu Barov
author: Jan J. Dijkstra
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: The authors deal with the phenomenon of nonconvex compact sets having convex projections (shadows) onto every hyperplane. They establish some properties of such sets and prove existence of such a set even with a prescribed condition. The main results are the following 2 theorems. Theorem 1. If \(C\) is a nonconvex compact set in the Euclidean \(n\)-space, \(n>2\), with all its shadows convex, then either (1) \(C\) contains an \((n-1)\)-cell (i.e. a subset homeomorphic to the standard closed \((n-1)\)-dimensional disc), or (2) there exist \(n+1\) distinct hyperplanes such that the intersection of \(C\) with any of them contains a topological \((n-2)\)-sphere. Theorem 4. For every \(n>2\), there exists a nonconvex compact set \(C\) in the Euclidean \(n\)-space with all its shadows convex such that there are precisely three distinct hyperplanes whose intersection with \(C\) contains a topological \((n-2)\)-sphere. (Application of Theorem 1 to the set \(C\) from Theorem 4 shows that \(C\) must contain an \((n-1)\)-cell.)
Mathematics Subject Classification ID: 52A99
zbMATH DE Number: 1848834
reviewed by: B. V. Dekster
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1848834