Porosity and diametrical completeness
type: Item
instance of: scholarly article
title: Porosity and diametrical completeness
zbMATH Open document ID: 1476.46021
published in: Israel Journal of Mathematics
publication date: 2021-07-12T00:00:00Z
review text: 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 \(X\).
reviewed by: Gilles Godefroy
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 7370917
zbMATH Keywords: porous sets
zbMATH Keywords: convex sets of constant width
zbMATH Keywords: normal structure
zbMATH Keywords: diametrically complete set
author: José Pedro Moreno
MaRDI profile type: Publication
full work available at URL: s11856-021-2151-z
OpenAlex ID: W3165286017
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DOI: 10.1007/S11856-021-2151-Z
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description: scientific article; zbMATH DE number 7370917