
Journal of Convex Analysis 26 (2019), No. 4, 13211336 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 solmazkh114@ipm.ir Mohamad R. Pouryayevali Dept. of Mathematics, University of Isfahan, Isfahan, Iran pourya@math.ui.ac.ir 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 d_{S} for a convex subset S in the cases where the boundary of S contains a geodesic segment, the boundary of S is C^{2} or the boundary of S is not regular is discussed. Furthermore, a nonsmooth version of positive semidefiniteness of the Hessian of convex functions on Riemannian manifolds is established. Keywords: Distance function, proximal normal cone, convexity, Riemannian manifold. MSC: 58C05, 53C21, 49J52 [ Fulltextpdf (150 KB)] for subscribers only. 