A New Upper Bound for the VC-Dimension of Visibility Regions

TL;DR

This paper proves that the VC-dimension of visibility regions in simple polygons is at most 14, improving understanding of geometric set systems and implications for guard placement strategies.

Contribution

It establishes a new upper bound of 14 for the VC-dimension of visibility regions in simple polygons, narrowing previous bounds significantly.

Findings

VC-dimension of visibility regions ≤ 14

Implication for fewer guards needed to cover polygons

Advances theoretical understanding of geometric visibility systems

Abstract

In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not "visually discernible", that is, T is not equal to the intersection of S with the visibility region vis(v) of any point v in P. In other words, the VC-dimension d of visibility regions in a simple polygon cannot exceed 14. Since Valtr proved in 1998 that d \in [6,23] holds, no progress has been made on this bound. By epsilon-net theorems our reduction immediately implies a smaller upper bound to the number of guards needed to cover P.