Hostname: page-component-54dcc4c588-m259h Total loading time: 0 Render date: 2025-09-11T23:39:05.828Z Has data issue: false hasContentIssue false
  • English
  • Français

ON CLOSED SETS WITH CONVEX PROJECTIONS

Published online by Cambridge University Press:  06 March 2002

STOYU BAROV ,
JOHN COBB  and
JAN J. DIJKSTRA
STOYU BAROV
Affiliation:
Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487-0350, USA Current address: Department of Mathematical Sciences, Ball State University, 465 Robert Bell Building, Muncie, IN 47306-0490, USA; stoyu@hotmail.com
JOHN COBB
Affiliation:
Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, USA; johncobb@uidaho.edu
JAN J. DIJKSTRA
Affiliation:
Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487-0350, USA Current address: Divisie der Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, Netherlands; dijkstra@cs.vu.nl
Get access

Abstract

A shadow of a subset A of ℝn is the image of A under a projection onto a hyperplane. Let C be a closed nonconvex set in ℝn such that the closures of all its shadows are convex. If, moreover, there are n independent directions such that the closures of the shadows of C in those directions are proper subsets of the respective hyperplanes then it is shown that C contains a copy of ℝn−2. Also for every closed convex set B ‘minimal imitations’ C of B are constructed, that is, closed subsets C of B that have the same shadows as B and that are minimal with respect to dimension.

Information

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 154 - 166
Copyright
2002 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable