The Motzkin theorem on nonpositively curved spaces
Let \(M\) denote a complete, simply connected \(n\)-dimensional Riemannian manifold with sectional curvature \(K\leq 0\). The space \(M\) satisfies many of the same convexity properties as flat Euclidean space \(\mathbb{R}^n\). In this article the author investigates generalizations related to the following Theorem (Motzkin). Let \(G\) be a closed subset of \(\mathbb{R}^n\) with the standard flat metric. Then the following properties are equivalent: 1) \(G\) is a convex set. 2) Each point \(x\) of \(\mathbb{R}^n\) has a unique point \(P(x)\) in \(G\) that is closest to \(x\). 3) The function \(d_G(x)= d(x,G)\) is differentiable on \(\mathbb{R}^n-G\). 4) The function \(d_G(x)= d(x,G)\) is \(C^{1+ \text{Lip}}\) on \(\mathbb{R}^n- G\). If \(M\) is as above, then there is a unique unit speed geodesic \(\gamma_{pq}\) joining any two distinct points \(p\), \(q\) of \(M\). Let \(v(p,q)\) denote the initial velocity at \(p\) of this geodesic. A set \(G\subseteq M\) is geodesically convex if it contains the unique geodesic segment \(\gamma_{pq}\) joining any two of its points \(p\), \(q\). Each unit vector \(v\) in TM defines a geodesic \(\gamma_v\) and an open horoball \(B(v)\), the union over all positive \(t\) of the open balls of radius \(t\) centered at \(\gamma_v t\). A closed set \(G\subseteq M\) is weakly convex if each point \(x\) of \(M-G\) has a unique point \(P(x)\) in \(G\) that is closest to \(x\) and the open horoball determined by \(v(Px,x)\) is disjoint from \(G\). Geodesically convex implies weakly convex. Examples of closed sets \(G\) that are weakly convex but not geodesically convex include the complements and boundaries of open horoballs in the case that \(M\) is not flat. The article proves two main results: Theorem 1. Let \(M\) be as above and suppose that \(-k^2\leq K\leq 0\) for some positive constant \(k\). Let \(G\) be a closed subset of \(M\). Then the following properties are equivalent: 1) \(G\) is weakly convex. 2) Each point \(x\) of \(M\) has a unique point \(P(x)\) in \(G\) that is closest to \(x\). 3) The function \(d_G(x)= d(x, G)\) is differentiable on \(M-G\). If any of these conditions is satisfied, then \(d_G(x)\) is \(C^1\) on \(M-G\) and the projection \(P: M\to G\) is locally Lipschitz. Theorem 2. Let \(G\) be a closed subset of a manifold \(M\) as above with \(K\leq 0\). Then the following properties are equivalent: 1) \(G\) is geodesically convex. 2) The function \(d_G(x)\) is \(C^1\) and the projection \(P: M\to G\) is \(1\)-Lipschitz.
scientific article; zbMATH DE number 1436014