Some geometry of convex bodies in C(K) spaces

We deal with some problems related to vector addition and diametric completion procedures of convex bodies in C(K) spaces. We prove that each of the following properties of convex bodies in C(K) characterizes the underlying compact Hausdorff space K as a Stonean space: (i) C(K) has a generating unit ball; (ii) all Maehara sets in C(K) are complete; (iii) the set Dd of all complete sets of diameter d in C(K) is convex. In contrast to these results, we further show the following for all C(K) spaces: (a) a weaker version of the generating unit ball property is satisfied; (b) there is a Maehara-like completion procedure; (c) Dd is starshaped with respect to any ball of radius d/2. The proofs are based on a systematic investigation of generalized order intervals and intersections of balls, which is carried out in the first part of the paper, and of diametrically complete sets. A new characterization of spaces `∞ among all n-dimensional real normed spaces, in terms of summands and intersections of translates of the unit ball, is finally provided. 2010 Mathematics Subject Classification: primary 52A05; secondary 46B20, 46E15, 52A21.