WINNING TACTICS IN A GEOMETRICAL GAME

A winning tactic for the point-closed slice game in a closed bounded convex set K with Radon-Nikodym property (RNP) is constructed. Consequently a Banach space X has the RNP if and only if there exists a winning tactic in the point-closed slice game played in the unit ball of X. By contrast, there is no winning tactic in the point-open slice game in K. Finally, a more subtle analysis of the properties of the winning tactics leads to a characterization of superreflexive spaces.