Slice-Continuous Sets in Reflexive Banach Spaces: Some Complements

In this paper we study a class of closed convex sets introduced recently by Ernst et al. (J Funct Anal 223:179–203, 2005) and called by these authors slice-continuous sets. This class, which plays an important role in the strong separation of convex sets, coincides in ℝ ⁿ with the well known class of continuous sets defined by Gale and Klee in the 1960s. In this article we achieve, in the setting of reflexive Banach spaces, two new characterizations of slice-continuous sets, similar to those provided for continuous sets in ℝ ⁿ by Gale and Klee. Thus, we prove that a slice-continuous set is precisely a closed and convex set which does not possess neither boundary rays, nor flat asymptotes of any dimension. Moreover, a slice-continuous set may also be characterized as being a closed and convex set of non-void interior for which the support function is continuous except at the origin. Dedicated to Boris Mordukhovich in honour of his 60th birthday.