
Journal of Convex Analysis 27 (2020), No. 4, 12611275 Copyright Heldermann Verlag 2020 A Riemannian Corollary of Helly's Theorem Alexander Rusciano Department of Mathematics, University of California, Berkeley, U.S.A. rusciano@math.berkeley.edu We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Grünbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least 1/(n+1) of the mass. As an application, the subgradient oracle complexity of convex optimization is polynomial in the size of the parameters defining the problem. Keywords: Helly's theorem, geodesic convexity, convex optimization. MSC: 52A01. [ Fulltextpdf (142 KB)] for subscribers only. 