Porosity and diametrical completeness (Q2040220)

This article is devoted to closed bounded convex subsets of Banach spaces which are diametrically complete, which means that they cannot be enlarged without increasing their diameter. Convex sets of constant width form a strict subclass of the class \(\mathcal{D}\) of diametrically complete sets. The purpose of this work is to show that \(\mathcal{D}\) is a rather small subset of the set \(\mathcal{H}\) of all closed bounded convex subsets of a Banach space \(X\) equipped with the Hausdorff distance. The relevant notions of smallness are metric conditions which imply in particular that the set is nowhere dense: a set \(M\) is uniformly very porous if every small enough open ball centered in \(x\in M\) contains a ball of fixed proportional diameter which is disjoint from \(M\). When the proportion is allowed to depend on \(x\in M\), then \(M\) is very porous. A Banach space \(X\) has Young constant \(J(X)<2\) if there is \(\lambda <1\) such that every bounded set \(C\) of diameter 1 is contained in some ball of radius \(\lambda\). The normal structure is a distinct but related notion where the ball is centered in some point of \(C\) but no uniformity on \(C\) is required, that is, \(\lambda=\lambda(C)\). The main result of the article is Theorem 3.1 which states that under the trivial restriction that \(X\) is not 1-dimensional, the set \(\mathcal{D}\) is closed and uniformly very porous if \(J(X)<2\), and, moreover, that \(\mathcal{D}\) is closed and very porous if \(X\) has normal structure. The proofs are geometrical and elementary, but quite involved. Unexpectedly, it seems to be unknown if the set \(\mathcal{D}\) is closed in \(\mathcal{H}\) for every normed space \(X\).