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 $\mathcal{A}$ is related to $K$ if each $A\in \mathcal{A}$ 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.

6 pages 2 figures

Cited by in corpus (1)

• On a problem by Dol'nikov