Abstract

We consider the set-valued mapping whose images are intersections of a fixed closed convex bounded set with nonempty interior from a real Hilbert space with shifts of a closed linear subspace. We characterize such strictly convex sets in the Hilbert space, that the considered set-valued mapping is Hölder continuous with the power 1/2 in the Hausdorff metric. We also consider the question about intersections of a fixed uniformly convex set with shifts of a closed linear subspace. We prove that the modulus of continuity of the set-valued mapping in this case is the inverse function to the modulus of uniform convexity and vice versa: the modulus of uniform convexity of the set is the inverse function to the modulus of continuity of the set-values mapping.

Author information

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

Maxim V. Balashov

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

balashov73@mail.ru

Keywords

  • Hilbert space
  • strongly convex set of radius R
  • Hausdorff metric
  • Lipschitz continuous set-valued mapping
  • Hölder continuous set-valued mapping
  • modulus of uniform convexity
  • modulus of continuity

Mathematics Subject Classification

49J52, 46C05, 26B25; 46B20, 52A07

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Suggested citation

M. V. Balashov. “Intersection of a Set with a Hyperplane.” Journal of Convex Analysis 23 (2016), No. 1, 227–236. https://doi.org/10.68381/jca23008

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press.