International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 2, Pages 267-273
doi:10.1155/S0161171285000291
A helly number for unions of two boxes in R2
Department of Mathematics, University of Oklahoma, Norman 73019, Oklahoma, USA
Received 13 February 1985
Copyright © 1985 Marilyn Breen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let S be a polygonal region in the plane with edges parallel to the coordinate axes. If every 5 or fewer boundary points of S can be partitioned into sets A and B so that conv A⋃ conv B⫅S, then S is a union of two convex sets, each a rectangle. The number 5 is best possible.
Without suitable hypothesis on edges of S, the theorem fails. Moreover, an example reveals that there is no finite Helly number which characterizes arbitrary unions of two convex sets, even for polygonal regions in the plane.