On Klee's problem of convex bodies in Banach spaces

It is well known that every convex body in a finite dimensional normed space can be uniformly approximated by strictly convex and smooth convex bodies. However, in the case of infinite dimensions, little progress has been made since Klee asked how it is in the case of infinite dimensions in 1959. In this paper, we show that for an infinite dimensional Banach space $X$, (1) every convex body can be uniformly approximated by strictly convex bodies if and only if $X$ admits an equivalent strictly convex norm; (2) every convex body can be uniformly approximated by G\^{a}teaux smooth convex bodies if the dual $X^*$ of $X$ admits an equivalent strictly convex dual norm; in particular, (3) if $X$ is either separable, or reflexive, then every convex body in $X$ can be uniformly approximated by strictly convex and smooth convex bodies. They are done by showing that some correspondences among the sets of all convex bodies endowed with the Hausdorff metric, all continuous coercive Minkowski functionals and Fenchel's transform defined on all quadratic homogenous continuous convex functions equipped with the metric induced by the sup-norm of all bounded continuous functions defined on the closed unit ball $B_X$ are actually locally Lipschitz isomorphisms.