Journal of the London Mathematical Society



ON CLOSED SETS WITH CONVEX PROJECTIONS


STOYU BAROV a1p1, JOHN COBB a2 and JAN J. DIJKSTRA a1p2
a1 Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487-0350, USA
a2 Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, USA; johncobb@uidaho.edu

Abstract

A shadow of a subset A of [open face R]n is the image of A under a projection onto a hyperplane. Let C be a closed nonconvex set in [open face R]n 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 [open face R]n−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.

(Received November 23 2000)
(Revised July 20 2001)


Correspondence:
p1 Current address: Department of Mathematical Sciences, Ball State University, 465 Robert Bell Building, Muncie, IN 47306-0490, USA; stoyu@hotmail.com
p2 Current address: Divisie der Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, Netherlands; dijkstra@cs.vu.nl