Journal of Convex Analysis 23 (2016), No. 1, 227--236
Copyright Heldermann Verlag 2016
Intersection of a Set with a Hyperplane
Maxim V. Balashov
Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny -- Moscow region, Russia 141700
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.
Keywords: Hilbert space, strongly convex set of radius R, Hausdorff metric, Lipschitz continuous set-valued mapping, Hoelder continuous set-valued mapping, modulus of uniform convexity, modulus of continuity.
MSC: 49J52, 46C05, 26B25; 46B20, 52A07
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