Continuous selections of Lipschitz extensions in metric spaces

Continuous Selections and Approximations in Α-convex Metric Spaces

In the paper, the notion of a generalized convexity was defined and studied from the view-point of the selection and approximation theory of set-valued maps. We study the simultaneous existence of continuous relative selections and graph-approximations of lower semicontinuous and upper semicontinuous set-valued maps with α-convex values having nonempty intersection.

Metric Embeddings and Lipschitz Extensions

Spaces of Lipschitz Functions on Metric Spaces

In this paper the universal properties of spaces of Lipschitz functions, defined over metric spaces, are investigated.

Lipschitz selections of the diametric completion mapping in Minkowski spaces

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Bückner in twoand three-dimensional Euclidean spaces. We prove that this generalized Bückner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces...

Infinitesimally Lipschitz Functions on Metric Spaces

For a metric space X, we study the space D∞(X) of bounded functions on X whose infinitesimal Lipschitz constant is uniformly bounded. D ∞(X) is compared with the space LIP∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D∞(X) with t...