Generalized derivatives of distance functions and the existence of nearest points
نویسندگان
چکیده
The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset G of X and x ∈ X \ G, the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 or−1 implying the existence of the nearest points to x from G is equivalent to X being compactly locally uniformly convex. Similar results for uniqueness of the nearest point are also established. © 2008 Elsevier Ltd. All rights reserved.
منابع مشابه
Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points
The relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or −1, then the ...
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