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Remotal sets revisited. (English) Zbl 1030.46018

Summary: Farthest point theory is not so rich and developed as nearest point theory, which has more applications. Farthest points are useful in studying the extremal structure of sets; see, e.g., the survey paper [T. D. Narang, Nieuw Arch. Wiskd., (3) 25, 54-79 (1977; Zbl 0342.46009)]. There are some interactions between the two theories; in particular, uniquely remotal sets in Hilbert spaces are related to the old open problem concerning the convexity of Chebyshev sets.
The aim of this paper is twofold: first, we indicate characterizations of inner product spaces and of infinite-dimensional Banach spaces, in terms of remotal points and uniquely remotal sets. Second, we try to update the survey paper [T. D. Narang, Nieuw Arch. Wiskd. 9, 1-12 (1991; Zbl 0779.54023)] concerning uniquely remotal sets.

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46C15 Characterizations of Hilbert spaces
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