Abstract

Developing the Polyak convexity principle, we show that strong convexity of a set in a Hilbert space is preserved under a C1,1C^{1,1} covering mapping provided that the parameters of the mapping and the set are related properly. We also prove that weak convexity of a set is preserved under C1,1C^{1,1} bi-Lipschitz mapping and obtain exact estimates of the convexity parameters of the images of the Cartesian product of two sets depending on the parameters of the sets and the mapping.

Author information

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

Grigorii E. Ivanov

Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia

g.e.ivanov@mail.ru

Keywords

  • C1
  • 1 mapping
  • covering mapping
  • bi-Lipschitz mapping
  • strong convexity
  • weak convexity
  • the Polyak convexity principle
  • Minkowski sum

Mathematics Subject Classification

52A05, 46T20

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Suggested citation

G. E. Ivanov. “Nonlinear Images of Sets. I: Strong and Weak Convexity.” Journal of Convex Analysis 27 (2020), No. 1, 361–380. https://doi.org/10.68381/jca27020

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.