Lipschitz retracts, selectors, and extensions
The authors show, under mild hypotheses, that the intersection of a Lipschitz continuous set-valued mapping and a Lipschitz continuous ball-valued mapping is again Lipschitz continuous. This allows to give simpler proofs, with improved Lipschitz constants, of several known results. In particular, the authors obtain: (1) the results of \textit{J. R. Isbell} [Pac. J. Math. 11, 609-648 (1961; Zbl 0109.15402)] that, for any Banach space \(X\), \({\mathcal H}(X)\) (i.e. the family of all closed, bounded, convex, nonempty subsets of \(X\), equipped with the Hausdorff metric) is the range of a Lipschitz retract on any metric space containing it; (2) a result of \textit{A. G. Skaletskij} [Mosc. Univ. Math. Bull. 40, No. 2, 29-53 (1985); translation from Vestn. Mosk. Univ., Ser. I 1985, No. 2, 24-28 (1985; Zbl 0609.54018)] implying that if \({\mathcal A}\) is a ``reasonable'' subset of \({\mathcal H}(X)\) with ``uniformly normal structure'' then there is a selector \(F:{\mathcal A}\to X\) whose restriction to each bounded subset of \({\mathcal H}(X)\) is uniformly continuous; (3) the result of \textit{A. Le Donne} and \textit{M. V. Marchi} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 68, 278-280 (1980; Zbl 0481.54011)] that there is a Lipschitz selector \(s:{\mathcal H}(\mathbb{R}^n) \to\mathbb{R}^n\) satisfying \(s(A_0)=x_0\) for any previously specified \(x_0\in A_0\in {\mathcal H}(\mathbb{R}^n)\); and (4) the generalization of this due to \textit{Z. Artstein} [Nonlinear Anal., Theory Methods Appl. 16, No. 7/8, 701-704 (1991; Zbl 0723.34007)], asserting that a Lipschitz selection on a subdomain can be extended to a Lipschitz selection on the entire domain of any set-valued mapping into \({\mathcal H} (\mathbb{R}^n)\).
Lipschitz continuous ball-valued mapping
scientific article; zbMATH DE number 881141