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Journal of Convex Analysis 27 (2020), No. 4, 1261--1275
Copyright Heldermann Verlag 2020

A Riemannian Corollary of Helly's Theorem

Alexander Rusciano
Department of Mathematics, University of California, Berkeley, U.S.A.

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.

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