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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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