Abstract

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 dSd_S for a convex subset S in the cases where the boundary of S contains a geodesic segment, the boundary of S is C2C^2 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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Distance function
  • proximal normal cone
  • convexity
  • Riemannian manifold

Mathematics Subject Classification

58C05, 53C21, 49J52

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Suggested citation

S. Khajehpour, M. R. Pouryayevali. “Convexity of the Distance Function to Convex Subsets of Riemannian Manifolds.” Journal of Convex Analysis 26 (2019), No. 4, 1321–1336. https://doi.org/10.68381/jca26069

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press.