Lipschitz extensions and approximations

Metric Embeddings and Lipschitz Extensions

Vector-valued Optimal Lipschitz Extensions

Consider a bounded open set U ⊂ Rn and a Lipschitz function g : ∂U → Rm. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variant...

Optimal Lipschitz Extensions and the Infinity Laplacian

We reconsider in this paper boundary value problems for the so-called “infinity Laplacian” PDE and the relationships with optimal Lipschitz extensions of the boundary data. We provide some fairly elegant new proofs, which clarify and simplify previous work, and in particular draw attention to the fact that solutions may be characterized by a comparison principle with appropriate cones. We in pa...

Lipschitz functions and fuzzy number approximations

We prove that some important properties of convex subsets of vector topological spaces remain valid in the space of fuzzy numbers endowed with the Euclidean distance. We use these results to obtain a characterization of fuzzy number-valued Lipschitz functions. This fact helps us to find the best Lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity intr...

Extensions of Lipschitz functions and Grothendieck's BAP

A metric compact space M is seen as the closure of the union of a sequence (Mn) of finite n-dense subsets. Extending to M (up to a vanishing uniform distance) Banach-space valued Lipschitz functions defined on Mn, or defining linear continuous near-extension operators for real-valued Lipschitz functions on Mn, uniformly on n is shown to be equivalent to the bounded approximation property for th...