Vol. 50, No. 1, 1974

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Intersectional properties of certain families of compact convex sets

B. E. Fullbright

Vol. 50 (1974), No. 1, 57–62

Let p and q be integers with p q 2. A family F of compact convex subsets of a finite dimensional linear space is said to have the (p,q)-property if F contains at least p sets and from each p sets of F some q have a common point. In this paper a family F is defined to have the (p,q,k)-property in a n-dimensional normed linear space if F has the (p,q)-property and an additional property which is measured by k, with 0 k 1. In some sense k measures the “squareness” of the members of F. The main result is that if k > 0, there exists a positive integer Pn(p,q,k) such that each family F with the (p,q,k)-property in a n-dimensional normed linear space can be partitioned into Pn(p,q,k) subfamilies each with a nonempty intersection.

Hadwiger and Debrunner have considered the following question: Is there a positive integer N(p,q,n) such that every finite family F of sets in En with the (p,q)-property can be partitioned into N(p,q,n) subfamilies each of which has a nonempty intersection?

Mathematical Subject Classification 2000
Primary: 52A20
Received: 26 September 1972
Revised: 18 December 1972
Published: 1 January 1974
B. E. Fullbright