On the geometry of locally nonconical convex sets
type: Item
instance of: scholarly article
title: On the geometry of locally nonconical convex sets
zbMATH Open document ID: 0937.52002
DOI: 10.1023/A:1005080830204
author: G. C. Shell
published in: Geometriae Dedicata
publication date: 2000-06-07T00:00:00Z
review text: If for every point \(x,y\) a convex subset \(Q\) of a Hausdorff topological vector space there exists a relative neighborhood \(U\) of \(x\) in \(Q\) such that \(U+{1\over 2}(y-x)\subset Q\), then we say that \(Q\) is locally nonconical. The author proves that a proper closed convex set \(Q\) with non-empty interior in a Hilbert space is locally nonconical if and only if for every line segment \([z,z'] \subset\text{bd}(Q)\) and for every \(p\in]z,z'[\), there exists a relative neighborhood \(U\) of \(p\) in \(Q\) such that the set \(U\cap \text{bd}(Q)\) is the union of a family of line segments parallel to \([z,z']\). It is shown that the closed unit ball of the Banach space of real sequences converging to 0 is locally nonconical.
Mathematics Subject Classification ID: 52A05
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 1295090
zbMATH Keywords: locally noncompact set
zbMATH Keywords: stable convex set
zbMATH Keywords: Hilbert space
reviewed by: Marek Lassak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1295090