Lion and Man game and fixed point free maps

This work is related with pursuit-evasion game where Lion is the pursuer and Man is the evader. We suppose that both players move in a metric space, have equal maximum speeds and complete information about the location of each other. We say that Man wins if he can escape a capture with non-zero radius; more precisely if there exists a positive number p and a non-anticipative strategy for some players' initial positions, that let him always be out of Lion's p-neighbourhood. We study sufficient conditions of the existence of Man's winning strategy. In this way we use the metric properties of space (mainly geodesics' behavior and fixed-point free maps). The technique requires neither convexity nor finite dimension of a space.