, Volume 152, Issue 3, pp 255-263
Date: 29 Aug 2007

Porosity and diametrically maximal sets in C(K)

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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 AD 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.