Vol. 69, No. 2, 1977

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Decompositions for nonclosed planar m-convex sets

Marilyn Breen

Vol. 69 (1977), No. 2, 317–324
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

σ(m) = (m  − 1)[1+ (2m−2 − 1)2m− 3].

Hence for m 3, S is expressible as a union of (m 1)32n3σ(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
Authors
Marilyn Breen