On strengthened versions of Klee's convex body problem in Banach spaces

Abstract

In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and G\^{a}teaux smoothness. In this paper, we continue to study this problem in strong senses, such as local uniform convexity, uniform convexity, Fr\'{e}chet smoothness and uniform smoothness. As a result, we show (1) Every convex body in a Banach space $X$ is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if $X$ admits an equivalent locally uniformly convex norm; (2) Every convex body in $X$ can be approximated by Fr\'echet smooth convex bodies if $X$ admits an equivalent norm so that its dual norm is locally uniformly convex on $X^*$; 3. Every convex body in $X$ can be approximated by both locally uniformly convex and Fr\'{e}chet smooth convex bodies if $X$ is reflexive; 4. If $X$ is separable, then every convex body in $X$ can be approximated by both locally uniformly convex and Fr\'{e}chet smooth convex bodies if and only if $X$ is an Asplund space; (5) the following statements are equivalent: A. $X$ is super reflexive; B. Every convex body in $X$ can be uniformly approximated by uniformly convex bodies; C. Every convex body in $X$ can be uniformly approximated by uniformly smooth convex bodies; D. Every convex body in $X$ can be uniformly approximated by both uniformly convex and uniformly smooth convex bodies.