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’.
This is a preview of subscription content, access via your institution.
References
Ancona, A.: Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature, Rev. Mat. Iberoamericana 10(1) (1994), 189–220.
Anderson, M. T.: The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), 701–721.
Ballmann,W.: Lectures on Spaces of Nonpositive Curvature, DMV Sem. 25, Birkhäuser, Basel, 1995.
Ballmann,W., Gromov, M. and Schroeder, V.: Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985.
Bangert, V. and Lang, U.: Trapping quasiminimizing submanifolds in spaces of negative curvature, Comment. Math. Helv. 71 (1996), 122–143.
Borbély, A.: A note on the Dirichlet problem at infinity for manifolds of negative curvature, Proc. Amer. Math. Soc. 114 (1992), 865–872.
Borbély, A.: Construction of convex sets in negatively curved manifolds, Proc. Amer. Math. Soc. 118 (1993), 205–210.
Borbély, A.: On the smoothness of the convex hull in negatively curved manifolds, J. Geom. 54(1995), 3–14.
Borbély, A.: Convexity at infinity and bounded harmonic functions, Bull. Austral. Math. Soc. 56 (1997), 63–68.
Borbély, A.: Some results on the convex hull of finitely many convex sets, Proc. Amer. Math. Soc. 126 (1998), 1515–1525.
Borbély, A.: The nonsolvability of the Dirichlet problem on negatively curved manifolds, Differential Geom. Appl. 8 (1998), 217–237.
Bowditch, B. H.: Notes on Gromov's hyperbolicity criterion for path-metric spaces, in É. Ghys, A. Haefliger and A. Verjovsky (eds), Group Theory from a Geometrical Viewpoint, World Scientific, Singapore, 1991, pp. 64–167.
Bowditch, B. H.: Some results on the geometry of convex hulls in manifolds of pinched negative curvature, Comment. Math. Helv. 69 (1994), 49–81.
Bridson, M. and Haefliger, A.: Metric Spaces of Non-Positive Curvature, in preparation.
Burago, Yu., Gromov, M. and Perel'man, G.: A.D. Alexandrov spaces with curvature bounded below, Russian Math. Surveys 47(2) (1992), 1–58.
Choi, H. I.: Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds, Trans. Amer. Math. Soc. 281 (1984), 691–716.
Eberlein, P. and O'Neill, B.: Visibility manifolds, Pacific J. Math. 46 (1973), 45–109.
Gromov, M.: Foliated Plateau problem, Part 1: Minimal varieties, Geom. Funct. Anal. 1 (1991), 14–79.
Gromov, M.: Hyperbolic groups, in S. M. Gersten (ed.), Essays in Group Theory, MSRI Publications 8, Springer-Verlag, New York, 1987, pp. 75–263.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hummel, C., Lang, U. & Schroeder, V. Convex Hulls in Singular Spaces of Negative Curvature. Annals of Global Analysis and Geometry 18, 191–204 (2000). https://doi.org/10.1023/A:1006698910715
Issue Date:
DOI: https://doi.org/10.1023/A:1006698910715