On Klee's problem of convex bodies in Banach spaces
Lixin Cheng, Chunlan Jiang, Liping Yuan
TL;DR
This work advances Klee’s infinite-dimensional convex-body problem by establishing precise norm-structure conditions under which convex bodies in a Banach space can be uniformly approximated by strictly convex or Gâteaux smooth bodies. The authors build a network of order-reversing, locally Lipschitz isomorphisms between cones of convex bodies and cones of quadratic convex functionals, mediated by Minkowski functionals and Fenchel transforms, and use inf-convolution to generate desired approximants. The main results show that density of strictly convex bodies in the convex-body cone $ rak C(X)$ is equivalent to the existence of an equivalent strictly convex norm on $X$, while density of Gâteaux smooth convex bodies requires an equivalent strictly convex dual norm on $X^*$. In particular, separable or reflexive spaces enjoy both densities, and a combined strictly convex and Gâteaux smooth approximation result holds when both the primal and dual norms have strict convexity, addressing long-standing facets of Klee’s problem in infinite dimensions and linking geometric convexity to norm-structure properties.
Abstract
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 Gâteaux 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.