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function result theorem function f relation lemma minimax proof nonempty definition space condition minimax result quasiconvex theory player problem vector strategy payoff function f semicontinuou supaa optimization topological inf pf class analysis point player 2 representation topological space minimax theorem l.s.c example affine player 1 section property quasiconvex functions optimization problem duality minimax theorems payoff lagrangian measure yield subset suppf borel optimization theory topology programming noncooperative games probability noncooperative equilibrium vector-valued supaa fe observation collection k-convexlike borel probability measures minimax results hausdorff space vectors x 1 quasiconvex analysis von neumann 1 i n quasiconvex function vector-valued function f equality application function k research b 0 b subsection separation result frenk consequence separation