Geodesic
Convexity of the Distance Function to Convex Subsets of Riemannian Manifolds
Journal of convex analysis, Tome 26 (2019) no. 4, pp. 1321-1336
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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.
Classification : 58C05, 53C21, 49J52
Mots-clés : Distance function, proximal normal cone, convexity, Riemannian manifold 
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     author = {S. Khajehpour and M. R. Pouryayevali},
     title = {Convexity of the {Distance} {Function} to {Convex} {Subsets} of {Riemannian} {Manifolds}},
     journal = {Journal of convex analysis},
     pages = {1321--1336},
     year = {2019},
     volume = {26},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2019_26_4_JCA_2019_26_4_a13/}
}
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S. Khajehpour; M. R. Pouryayevali. Convexity of the Distance Function to Convex Subsets of Riemannian Manifolds. Journal of convex analysis, Tome 26 (2019) no. 4, pp. 1321-1336. http://geodesic.mathdoc.fr/item/JCA_2019_26_4_JCA_2019_26_4_a13/