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.
Pursuit-evasion game Lion-and-man game Simple pursuit Betweenness Convex metrics
Mathematics Subject Classification
91A23 91A24 49N75 53C22
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