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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
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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.

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

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