Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary

TL;DR

This paper analyzes pursuit-evasion games in specific geometric spaces, proving that one pursuer can win in CAT(0) spaces and three pursuers can win in certain compact Euclidean domains with analytic boundaries.

Contribution

It extends pursuit-evasion theory by establishing winning strategies for pursuers in CAT(0) spaces and in compact Euclidean domains with piecewise analytic boundaries.

Findings

One pursuer has a winning strategy in any CAT(0) space.

Three pursuers can always win in compact Euclidean domains with piecewise analytic boundary.

Extends previous work from polygonal environments to more general settings.

Abstract

In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game. We consider the most restictive capture condition: a pursuer must become colocated with the evader to win the game. We prove two general results about pursuit-evasion games in topological spaces. First, we show that one pursuer has a winning strategy in any CAT(0) space under this restrictive capture criterion. This complements a result of Alexander, Bishop and Ghrist, who provide a winning strategy for a game with positive capture radius. Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic. We show that three pursuers always have a winning strategy by extending recent work of Bhadauria, Klein, Isler and Suri from polygonal environments to our more general setting.