Abstract

We compare four generalizations of convex sets which ensure good properties of the projection: p-convexity (widely identical to ρ-prox regularity), proximal smoothness, quasi-convexity and strict quasi-convexity, and one generalization based on the distance function: approximate convexity. We prove that (i) p-convex sets essentially coincide with quasi-convex sets, (ii) strictly quasi-convex sets are a subclass of proximally smooth sets and (iii) p-convex or quasi-convex sets are approximately convex, but the converse is false. The definition and main properties of these approaches are recalled without demonstration but with unified notations. We compare the Lipschitz properties of the projection on the different families of sets, and show that strict quasi-convexity ensures moreover the unimodality of the distance to a point over the set, and hence the computability of the projection by local optimization algorithms. Sufficient size×\timescurvature conditions for strict quasi-convexity are also recalled.

Author information

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

Guy Chavent

Ceremade, Université Paris-Dauphine, 75775 Paris Cedex 16, France
and: Inria-Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France

guy.chavent@free.fr

Keywords

  • Generalized set convexity
  • p-convexity
  • prox-regularity
  • approximate convexity
  • proximal smoothness
  • quasi-convexity
  • strict quasi-convexity
  • geodesics
  • single-valued projection
  • closest points
  • Hilbert space

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Suggested citation

G. Chavent. “On p-Convex, Proximally Smooth, Quasi-Convex, Strictly Quasi-Convex and Approximately Convex Sets.” Journal of Convex Analysis 22 (2015), No. 2, 427–446. https://doi.org/10.68381/jca22019

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press.