A characterization theorem for compact unions of two starshaped sets in R³

Marilyn Breen

Vol. 128 (1987), No. 1, 63–72

DOI: 10.2140/pjm.1987.128.63

Abstract

Set S in R^d has property P_k if and only if S is a finite union of d-polytopes and for every finite set F in bdry S there exist points c₁,…,c_k (depending on F) such that each point of F is clearly visible via S from at least one c_i, 1 ≤ i ≤ k. The following results are established.

1. Let S ⊆ R³. If S satisfies property P₂, then S is a union of two starshaped sets.
2. Let S ⊆ R^d, d ≥ 3. If S is a compact union of k starshaped sets, then there exists a sequence {S_j} converging to S (relative to the Hausdorff metric) such that each set S_j satisfies property P_k.

When d = 3 and k = 2, the converse of (2) above holds as well, yielding a characterization theorem for compact unions of two starshaped sets in R³.

Mathematical Subject Classification 2000

Primary: 52A30

Secondary: 52A25

Milestones

Received: 21 November 1985

Revised: 21 August 1986

Published: 1 May 1987