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Porosity and diametrically maximal sets in C(K)
Authors:J P Moreno
Institution:(1) Universidad Autónoma de Madrid, Madrid, Spain
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 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.
Keywords:2000 Mathematics Subject Classification: 54E52  46B20
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