Approximate Capture in Gromov-Hausdorff Closed Spaces

We consider the Lion and Man game, i.e., a two-person pursuit-evasion game with equal players' top speeds. We assume that capture radius is positive and chosen in advance. The main aim of the paper is describing pursuer's winning strategies in general compact metric spaces that are close to the given one in the sense of Gromov-Hausdorff distance. We prove that the existence of $\alpha$-capture by a time $T$ in one compact geodesic space implies the existence of $(\alpha + (20T +8)\sqrt{\varepsilon})$-capture by this time $T$ in any compact geodesic space that is $\varepsilon$-close to the given space. It means that capture radii (in a nearby spaces) tends to the given one as the distance between spaces tends to zero. Thus, this result justifies calculations on graphs instead of complicated spaces.