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Journal of Convex Analysis 26 (2019), No. 4, 1321--1336
Copyright Heldermann Verlag 2019

Convexity of the Distance Function to Convex Subsets of Riemannian Manifolds

Solmaz Khajehpour
School of Mathematics, Institute for Research in Fundamental Sciences, Tehran, Iran

Mohamad R. Pouryayevali
Dept. of Mathematics, University of Isfahan, Isfahan, Iran

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function dS for a convex subset S in the cases where the boundary of S contains a geodesic segment, the boundary of S is C2 or the boundary of S is not regular is discussed. Furthermore, a nonsmooth version of positive semi-definiteness of the Hessian of convex functions on Riemannian manifolds is established.

Keywords: Distance function, proximal normal cone, convexity, Riemannian manifold.

MSC: 58C05, 53C21, 49J52

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