Upper bounds for the piercing number of families of pairwise intersecting convex polygons

Details Upper bounds for the piercing number of families of pairwise intersecting convex polygons Upper bounds for the piercing number of families of pairwise intersecting convex polygons

2011-12-31

arxiv.org/abs/1201.0241v2

Mathematics

Metric Geometry

6 pages 2 figures

Scientific paper

A convex polygon $A$ is related to a convex $m$-gon $K= \bigcap_{i=1}^m k_i^+$, where $k_1^+,..., k_m^+$ are the $m$ halfplanes whose intersection is equal to $K$, if $A$ is the intersection of halfplanes $a_1^+,...,a_l$, each of which is a translate of one of the $k_i^+$-s. The planar family ${\cal A}$ is related to $K$ if each $A \in {\cal A}$ is related to $K$. We prove that any family of pairwise intersecting convex sets related to a given $n$-gon has a finite piercing number which depends on $n$. In the general case we show $O(3^{n^3})$, while for a certain class of families, we decrease the bound to $4(n-2)$, and for $n=3,4$ the bound is 3 and 6 respectively.

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