Abstract

We prove that the function which for a given point of a Banach space gives the largest distance to the points of a given convex closed bounded set (antidistance) is weakly concave on the complement to some neighborhood of the set if and only if the set is a summand of some ball of some radius. We obtain precise estimates for parameters of weak concavity via the size of the neighborhood and radius of the ball in the Hilbert space.

Author information

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

Maxim V. Balashov

Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow region, Russia 141700

balashov73@mail.ru

Maxim O. Golubev

Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow region, Russia 141700

maksimkane@mail.ru

Keywords

  • Antidistance function
  • uniform convexity
  • uniform smoothness
  • weak concavity
  • modulus of weak concavity
  • proximal smoothness
  • P-supporting condition

Mathematics Subject Classification

49J52, 58C20, 52A07; 26B25, 52A41, 52A05

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Suggested citation

M. V. Balashov, M. O. Golubev. “Weak Concavity of the Antidistance Function.” Journal of Convex Analysis 21 (2014), No. 4, 951–964. https://doi.org/10.68381/jca21051

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.