Lion and Man Game in Compact Spaces

TL;DR

This paper investigates pursuit-evasion games in compact geodesic spaces, establishing conditions under which the pursuer (Lion) can always guarantee a win using simple pursuit strategies, regardless of initial positions.

Contribution

It proves that simple pursuit strategies guarantee a win for the pursuer in broad classes of compact geodesic spaces satisfying the betweenness property, without requiring smoothness or finite dimension.

Findings

Lion wins in compact CAT(0)-spaces and Ptolemy spaces.

Simple pursuit strategy is sufficient for guaranteed capture.

No need for smoothness or boundary regularity assumptions.

Abstract

The pursuit-evasion game with two persons is considered. Both players are moving in a metric space, have equal maximum speeds and complete information about the location of each other. We study the sufficient conditions for a capture (with a positive capture radius). We assume that Lion wins if he manages the capture independently of the initial positions of the players and the evader's strategy. We prove that the discrete-time simple pursuit strategy is a Lion's winning strategy in a compact geodesic space satisfying the betweenness property. In particular, it means that Lion wins in compact CAT(0)-spaces, Ptolemy spaces, Buseman convex spaces or any geodesic space with convex metric. We also do not need to use such properties as finite dimension, smoothness, boundary regularity or contractibility of the loops.