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Convex hulls in singular spaces of negative curvature


Hummel, C; Lang, U; Schroeder, V (2000). Convex hulls in singular spaces of negative curvature. Annals of Global Analysis and Geometry, 18(2):191-204.

Abstract

The paper gives a simple example of a complete CAT(–1)-space containing a set S with the following property: the boundary at infinity ∂ ∞CH(S)of the convex hull of S differs from S by an isolated point. In contrast to this it is shown that if S is a union of finitely many convex subsets of a complete CAT(–1)-space X, then ∂ ∞CH(S) = ∂ ∞ S. Moreover, this identity holds without restrictions on S if CH is replaced by some notion of 'almost convex hull'.

Abstract

The paper gives a simple example of a complete CAT(–1)-space containing a set S with the following property: the boundary at infinity ∂ ∞CH(S)of the convex hull of S differs from S by an isolated point. In contrast to this it is shown that if S is a union of finitely many convex subsets of a complete CAT(–1)-space X, then ∂ ∞CH(S) = ∂ ∞ S. Moreover, this identity holds without restrictions on S if CH is replaced by some notion of 'almost convex hull'.

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5 citations in Web of Science®
3 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Alexandrov space; almost convex; convex hull; negative curvature
Language:English
Date:2000
Deposited On:29 Nov 2010 16:27
Last Modified:05 Apr 2016 13:26
Publisher:Springer
ISSN:0232-704X
Publisher DOI:https://doi.org/10.1023/A:1006698910715
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1744590
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0993.53012

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