Convex hulls in singular spaces of negative curvature
type: Item
instance of: scholarly article
title: Convex hulls in singular spaces of negative curvature
zbMATH Open document ID: 0993.53012
DOI: 10.1023/A:1006698910715
author: Christoph Hummel
author: Urs Lang
author: Viktor Schroeder
published in: Annals of Global Analysis and Geometry
publication date: 2002-09-23T00:00:00Z
review text: The paper contains three results related to the convex hulls of unbounded subsets in singular spaces of negative curvature. Theorem 1 gives a simple example of a complete \(\text{CAT}(-1)\) space \(X\) containing a set \(S\) such that the boundary at infinity \(\partial_{\infty}CH(S)\) of the convex hull of \(S\) differs from \(\partial_{\infty}S\) by an isolated point. The space \(X\) is a so-called parabolic cone over a (nonlocally compact) tree. In contrast to this, the authors show that for a locally compact \(X\), \(\partial_{\infty}CH(S)\) is always connected to \(\partial_{\infty}S\). The example from Theorem 1 is motivated by examples due to \textit{A. Ancona} [Rev. Mat. Iberoam. 10, No. 1, 189-220 (1994; Zbl 0804.58056)] and \textit{A. Borbély} [Differ. Geom. Appl. 8, No. 3, 217-237 (1998; Zbl 0947.53019)] of Hadamard manifolds with an upper curvature bound \(K\leq -1\) having similar properties. The last examples are however significantly more complicated. The study of the convex hulls of unbounded subsets in \(X\) is related to the Dirichlet problem at infinity since the property \(\partial_{\infty}CH(S)=\partial_{\infty}S\) for each \(S\subset X\) leads to the existence of bounded solutions of that problem [\textit{M. T. Anderson}, J. Differ. Geom. 18, 701-721 (1983; Zbl 0541.53036)]. The second result of the paper, Theorem 2, says that \(\partial_{\infty}CH(S)=\partial_{\infty}S\) for each \(S\) which is the union of finitely many convex subsets in a complete \(\text{CAT}(-1)\) space \(X\). This is a generalization to the singular setting of a result of \textit{A. Borbély} [Bull. Aust. Math. Soc. 56, No. 1, 63-68 (1997; Zbl 0884.53028)] proved in the Riemannian case. Finally, by Theorem 3, \(\partial_{\infty}CH_a(S)=\partial_{\infty}S\) for each subset \(S\) in a complete \(\text{CAT}(-1)\) space \(X\) and for each \(a>0\). Here \(CH_a(S)\) is the \(a\)-convex hull of \(S\), the notion introduced by the authors; \(CH_a(S)\subset CH(S)\) and \(CH(S)=\big…
reviewed by: Sergei V. Buyalo
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1436920
zbMATH Keywords: Alexandrov space
zbMATH Keywords: almost convex
zbMATH Keywords: convex hull
zbMATH Keywords: negative curvature
MaRDI profile type: Publication
full work available at URL: a:1006698910715
OpenAlex ID: W145318498
description: scientific article; zbMATH DE number 1436920