Piercing Pairwise Intersecting Geodesic Disks by Five Points

TL;DR

This paper proves that five points are always enough to intersect all pairwise intersecting geodesic disks within a simple polygon, improving previous bounds from 14 to 5.

Contribution

The authors establish a new, tighter bound of five points for piercing all pairwise intersecting geodesic disks in a simple polygon, advancing geometric piercing theory.

Findings

Five points suffice to pierce all disks

Improved the bound from 14 to 5 points

Applicable to geodesic disks in polygons

Abstract

Given a simple polygon $P$ on $n$ vertices, and a set $D$ of $m$ pairwise intersecting geodesic disks in $P$, we show that five points in $P$ are always sufficient to pierce all the disks in $D$. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer.