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NEW RESULTS FOR T(k)-FAMILIES IN THE PLANE

Part of: General convexity

Published online by Cambridge University Press:  10 December 2009

Andreas F. Holmsen
Andreas F. Holmsen*
Affiliation:
Department of Mathematical Sciences, KAIST, 335 Gwahangno (373-1 Guseong-dong), Yuseong-gu, Daejeon 305-701, South Korea (email: [email protected])
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Abstract

A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F. In this paper we study some basic properties of T(k)-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T(k)-family admits a partial transversal of size α∣F∣ for some constant α(k) which is independent of F. Here it will be shown that (2/(k(k−1)))1/(k−2)≤α(k)≤((k−2)/(k−1)), which are the best bounds to date.

MSC classification

Secondary: 52A35: Helly-type theorems and geometric transversal theory
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 86 - 92
Copyright
Copyright © University College London 2010

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