Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points

Authors: Li C.¹; Ni R.²

Source: Journal of Approximation Theory, Volume 115, Number 1, March 2002 , pp. 44-55(12)

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Abstract:

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 generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc. 39, 233–238). © 2002 Elsevier Science (USA).

Language: English

Document Type: Research article

Affiliations: 1: Department of Applied Mathematics, Southeast University, Nanjing, 210096, People's Republic of China 2: Department of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, 321000, People's Republic of China

Publication date: 2002-03-01