Derivatives of generalized distance functions and existence of generalized nearest points
type: Item
instance of: scholarly article
title: Derivatives of generalized distance functions and existence of generalized nearest points
zbMATH Open document ID: 1006.46015
author: C. Li
author: Renxing Ni
published in: Journal of Approximation Theory
publication date: 2003-03-12T00:00:00Z
review text: Let \(X\) be a real Banach space of dimension at least 2, \(C\) a closed bounded convex subset of \(X\) with \(0\in\text{int }C\). The Minkowski functional \(p_C: X\to \mathbb{R}\) with respect to the set \(C\) is defined by \[ p_C(x)= \inf\{\alpha> 0: x\in\lambda C\},\quad \forall x\in X. \] For a closed nonempty subset \(G\) of \(X\), the generalized distance function is defined by \(d_G(x)= \inf_{z\in G} p_C(x- z)\), \(\forall x\in X\). A point \(z_0\in G\) with \(p_C(x- z_0)= d_G(x)\) is called a generalized nearest point to \(x\) from \(G\). For any \(x,y\in X\), the one-sided directional derivative of \(d_G\) at \(x\) is \[ d_G'(x)(y)= \lim_{t\to 0^+} {d_G(x+ ty)- d_G(x)\over t}. \] In this paper, the authors investigate the relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces. It is proved that if the one-sided diretional derivative of the generalized distance function associated to \(G\) at \(x\) equals to \(1\) or \(-1\), then the generalized nearest point to \(x\) from \(G\) exists, provided that \(X\) is a compactly locally uniformly convex Banach space. Moreover, a partial answer to the following open problem put forward by \textit{S. Fitzpatrick} [Bull. Austr. Math. Soc. 39, No. 2, 233-238 (1989; Zbl 0674.46011)] is given. If \(G\) is a closed subset of reflexive Banach space \(X\), is the set \[ D= \{x\in X\mid G; \exists y\in \text{Bd}(c)\text{ with }d_G'(x)(y)= 1\} \] residual in \(X\setminus G\)? Theorem 3.5 in the present paper says that the answer to the problem is affirmative if \(C\) is both strictly convex and Kadeč. Whether Theorem 3.5 remains true without this assumption remains not known.
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1750339
zbMATH Keywords: compactly locally uniformly convex
zbMATH Keywords: strictly convex and Kadeč set
zbMATH Keywords: Minkowski functional
zbMATH Keywords: generalized distance function
zbMATH Keywords: generalized nearest point
zbMATH Keywords: directional derivatives
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: jath.2001.3651
OpenAlex ID: W2101116900
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DOI: 10.1006/JATH.2001.3651
description: scientific article; zbMATH DE number 1750339