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
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 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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