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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?
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