A proof of the Markov-Kakutani fixed point theorem via the Hahn-Banach theorem.

Dirk Werner

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On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals. ...

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CONTENTSIntroduction............................................................................................................................................................................... 5§ 1. Finite systems of convex inequalities.......................................................................................................................... 6§ 2. Infinite systems of convex inequalities...........................................................................................................................

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We study different aspects of the representation of weak*-compact convex sets of the bidual X** of a separable Banach space X via a nested sequence of closed convex bounded sets of X.

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