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A short proof of an interesting Hellytype Nina Amenta \Lambda

Summary: A short proof of an interesting Helly­type
Nina Amenta \Lambda
The Geometry Center
1300 South Second Street
Minneapolis, MN 55454
We give a short proof of the theorem that any family of subsets
of R d , with the property that the intersection of any non­empty finite
subfamily can be represented as the disjoint union of at most k closed
convex sets, has Helly number at most k(d + 1).
1 Introduction
We say that a family of sets F has Helly number h when h is the smallest
integer (if one exists) such that any finite subfamily H ` F has nonempty
intersection if and only if every subfamily B ` H with jBj Ÿ h also has
nonempty intersection. Theorems of the form ``F has Helly number h'' are
called Helly­type theorems -- they follow the model of Helly's theorem, which
states that the family of convex sets in R d has Helly number d+ 1. There are
many Helly­type theorems; for excellent surveys see [DGK63] and the recent


Source: Amenta, Nina - Department of Computer Science, University of California, Davis


Collections: Biology and Medicine; Computer Technologies and Information Sciences