Decompositions for nonclosed planar m-convex sets

Marilyn Breen

Vol. 69 (1977), No. 2, 317–324

DOI: 10.2140/pjm.1977.69.317

Abstract

Let S be an m-convex set in the plane having the property that (int cl S) ∼ S contains no isolated points. If T is an m-convex subset of S having convex closure, then T is a union of σ(m) or fewer convex sets, where

Hence for m ≥ 3, S is expressible as a union of (m − 1)³2ⁿ⁻³σ(m) or fewer convex sets.

In case S is m-convex and (int cl S) ∼ S contains isolated points, an example shows that no such decomposition theorem is possible.

Mathematical Subject Classification 2000

Primary: 52A10

Milestones

Received: 1 May 1975

Revised: 29 October 1975

Published: 1 April 1977