Abstract
For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.
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Bibliography
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis, Fizmatlit, Moscow, 2004.
J. Diestel, Geometry of Banach Spaces, Lecture Notes in Math, 485, Springer, Berlin, 1975; Russian transl.: Vishcha Shkola, Kiev, 1980.
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Translated from Matematicheskie Zametki, vol. 80, no. 2, 2006, pp. 163–171.
Original Russian Text Copyright © 2006 by M. V. Balashov, G. E. Ivanov.
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Balashov, M.V., Ivanov, G.E. On farthest points of sets. Math Notes 80, 159–166 (2006). https://doi.org/10.1007/s11006-006-0123-6
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DOI: https://doi.org/10.1007/s11006-006-0123-6