A note on reflexivity and nonconvexity (Q642552)

The Eisenfeld-Lakshmikantham (E-L, for short) measure of nonconvexity of a bounded subset of \(X\) is defined by \[ \mu(A)= \sup_{x\in\overline{\text{conv}(A)}} \underset{a\in A}{}{\text{inf}}\,\| x-a\|= H(A,\overline{\text{conv}(A))}), \] where \(H(C, D)\) is the Hausdorff-Pompeiu distance between the bounded subsets \(C\) and \(D\) of \(X\) (see [\textit{J. Eisenfeld} and \textit{V. Lakshmikantham}, Yokohama Math. J. 24, 133--140 (1976; Zbl 0361.34051)]). In the present paper, the author proves the following theorem. Theorem 2.5. For a Banach space \(X\), the following statements are equivalent: (i) \(X\) is reflexive; (ii) the E-L measure of nonconvexity \(\mu\) in \(X\) satisfies the Cantor property (i.e., for every decreasing sequence \(\{A_n\}^\infty_{n=1}\) of nonempty closed and bounded subsets of \(X\) such that \(\lim_{n\to\infty}\mu(A_n)= 0\), the closed and bounded set \(A_\infty= \bigcap^\infty_{n=1} A_n\) is nonempty and convex). Using this characterization, some results in best approximation and fixed point theory for reflexive Banach space are generalized by removing convexity requirements.