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

Abstract

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.