Exposed points and convexity (Q1803516)

Let \(M\) be a complete simply connected manifold without focal points, i.e. all distance balls are strictly convex. Many statements on convex subsets of \(\mathbb{R}^ n\) have an obvious generalization to this situation. The horospheres (horodisks) play the rôle of the hyperplanes (halfspaces). A point \(p\) in some subset \(B\subset M\) is called exposed point if \(B\) is contained in a supporting horodisk \(H\) with \(\partial H\cap B=\{p\}\). The two main results of the paper are: (1) Any compact subset \(B\subset M\) has at least two exposed points. (Proof: \(B\) lies in the intersection of two balls of radius \(\lambda=\text{diam}(B)\) which support \(B\) at two farthest points.) (2) A compact hypersurface \(S\) containing only exposed points bounds a strictly convex subset \(B\). In fact, \(B\) is the intersection of all its supporting horodisks.