An intersection property for starshaped sets in the plane

✍ Scribed by Marilyn Breen

• Publisher: Springer
• Year: 1991
• Tongue: English
• Weight: 451 KB
• Volume: 37
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00181408

✦ Synopsis

Let ~-be a family of compact starshaped sets in the plane. If every three and every two members of ~ have a union which is connected and simply connected, then N {F: F in ~} is simply connected and nonempty. Of course, if every three and every two members of ~-have a starshaped union, the same result holds.

1. Introduction

We begin with some familiar definitions. Let S be a subset of R d. For points x and y in S, we say x sees y via S if and only if the associated segment [x, 3:] lies in S. Set S is called starshaped if and only if there is some point p in S such that p sees via S each point of S, and the collection of all such points p is the (convex) kernel of S.

Various types of combinatorial results have been established for starshaped sets. There is a well-known characterization theorem by