Intersection of a Set with a Hyperplane

M. V. Balashov

Journal of convex analysis, Tome 23 (2016) no. 1, pp. 227-236

Résumé

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.

Détail

Classification : 49J52, 46C05, 26B25, 46B20, 52A07
Mots-clés : 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