The Lion and Man Game on Polyhedral Surfaces with Boundary

Abstract

We study the lion-and-man game in which a group of lions (the pursuers) try to capture a man (the evader). The players have equal speed. They can observe each other at all times. While the game is well-studied in two dimensional domains such as polygons, very little is known about its properties in higher dimensions. In this paper, we study the lion and man game when played on the surface of a three-dimensional solid represented as a polyhedron with boundary. We show that three lions with non-zero capture distance $delta$ can capture the man in finite time $O((frac{A}{delta^2} + frac{L}{delta})^2 frac{delta}{2})$ where $A$ is the area of the surface, and $L$ is the total edge length of the surface.