The Motzkin theorem on nonpositively curved spaces
type: Item
instance of: scholarly article
title: The Motzkin theorem on nonpositively curved spaces
zbMATH Open document ID: 0968.53027
DOI: 10.1023/A:1005236325541
author: Stéphane Grognet
published in: Geometriae Dedicata
publication date: 2001-09-11T00:00:00Z
review text: 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 \(M\) ha…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1436014
zbMATH Keywords: Motzkin's theorem
zbMATH Keywords: negative curvature
zbMATH Keywords: convexity
zbMATH Keywords: convex set
zbMATH Keywords: geodesically convex
zbMATH Keywords: weakly convex
zbMATH Keywords: horoball
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
full work available at URL: a:1005236325541
OpenAlex ID: W2913446882
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description: scientific article; zbMATH DE number 1436014