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

TL;DR

This paper establishes upper bounds on the piercing number for families of pairwise intersecting convex polygons related to a fixed polygon, with bounds depending on the number of sides of the polygon.

Contribution

It introduces bounds on the piercing number for such families, improving understanding of intersection properties of convex polygons related to a fixed shape.

Findings

Finite piercing number depending on polygon sides

General bound of O(3^{n^3}) for related convex sets

Reduced bounds for specific classes, e.g., 4(n-2)

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.