Porosity and diametrically maximal sets in C(K)

Abstract.

This paper is motivated by recent attempts to investigate classical notions from finite-dimensional convex geometry in spaces of continuous functions. Let ${\cal H}$ be the family of all closed, convex and bounded subsets of C(K) endowed with the Hausdorff metric. A completion of $A \in {\cal H}$ is a diametrically maximal set $D \in {\cal H}$ satisfying A ⊂ D and diam A = diam D. Using properties of semicontinuous functions and an earlier result by Papini, Phelps and the author [12], we characterize the family γ(A) of all possible completions of $A\in{\cal H}$. We give also a formula which calculates diam γ(A) and prove finally that, if K is a Hausdorff compact space with card K > 1, then the family of those elements of ${\cal H}$ having a unique completion is uniformly very porous in ${\cal H}$ with a constant of lower porosity greater than or equal to 1/3.