Abstract

A subset X of a real Hilbert space H is said to be proximally smooth provided that the function dX:H→Rd_X : H \to R (the distance to X) is continuously differentiable on an open tube U around X. It is proven that this property is equivalent to dXd_X having a nonempty proximal subgradient at every point of U, and that the (Gâteaux = Fréchet) derivative is locally Lipschitz on U. The Lipschitz behavior of the derivative is a consequence of the fact that under proximal smoothness, the metric projection onto X is single valued and Lipschitz on U. Alternate characterizations of proximal smoothness are given as well, in terms of properties of the proximal normal cone multifunction on X and on nearby closed neighborhoods of X. In case X is weakly closed, the list of equivalences is extended to include each point of U admitting a unique closest point in X. Further specializations are given in finite dimensions. In that setting, we discuss properties of locally Lipschitz real valued functions whose epigraphs are proximally smooth in a local sense. It is demonstrated that this function class coincides with the lower–C2C^2 functions studied by Rockafellar.

Author information

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

F. H. Clarke

Centre de recherches mathématiques, Université de Montréal, Montréal, Québec H3C 3J7, Canada.

R. J. Stern

Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H4B 1R6, Canada.

P. R. Wolenski

Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803, USA.

Keywords

  • Hilbert space
  • distance function
  • proximal smoothness
  • Gâteaux
  • strict and Fréchet derivatives
  • closest points
  • nonsmooth analysis
  • optimization
  • epigraph
  • lower–C² function

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Suggested citation

F. H. Clarke, R. J. Stern, P. R. Wolenski. “Proximal Smoothness and the Lower-C^2 Property.” Journal of Convex Analysis 2 (1995), No. 1&2, 117–144. https://doi.org/10.68381/jca02008

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Published by Heldermann Verlag, 1995. Rights now held by Banach Press LLC.