Convex hulls in singular spaces of negative curvature
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 \(CH(S)=\bigcup_{a>0}CH_a(S)\). It remains unclear whether there is a relation between the result of Theorem 3 and the Dirichlet problem at infinity.
scientific article; zbMATH DE number 1436920