Directionally Pseudo-Lipschitz Set-Valued Mappings

The Whitney Extension Problem and Lipschitz Selections of Set-valued Mappings in Jet-spaces

We study a variant of the Whitney extension problem (1934) for the space Ck,ω(Rn). We identify Ck,ω(Rn) with a space of Lipschitzmappings from Rn into the space Pk × Rn of polynomial fields on Rn equipped with a certain metric. This identification allows us to reformulate the Whitney problem for Ck,ω(Rn) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets o...

Absolutely Minimal Lipschitz Extension of Tree-valued Mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics — a generalization of a game called Tug of War that has been used in [42] to study real-valued AMLEs.

Weighted Variational Inequalities with Set-valued Mappings

Because of their applications in economics, game theory, mathematical physics, operations research and other areas, many classes of vector variational inequalities were intensively studied. For existence of solutions, resolution methods or equivalence with equilibrium and optimization problems see, for example, [11], [14], [15], [19], [17] and the references therein. For the study of variationa...

A Sard Theorem for Set-Valued Mappings∗†

If F is a set-valued mapping from IRn into IRm with closed graph, then y ∈ IRm is a critical value of F if for some x with y ∈ F (x), F is not metrically regular at (x, y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m − 1 (resp. a ...

Fixed Point Theorems for Set-valued Mappings