Derivatives of generalized distance functions and existence of generalized nearest points
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.
scientific article; zbMATH DE number 1750339
scientific article