Mathematical Notes

, Volume 80, Issue 3–4, pp 461–467 | Cite as

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

  • M. V. Balashov
  • G. E. Ivanov


We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.

Key words

weakly convex sets metric projection set-valued mapping modulus of continuity Chebyshev layer Hausdorff distance 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • M. V. Balashov
    • 1
  • G. E. Ivanov
    • 1
  1. 1.Moscow Institute of Physics and TechnologyMoscow

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