Convex hulls in singular spaces of negative curvature

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'.