Continuity of extensions of Lipschitz maps

Abstract

We establish the sharp rate of continuity of extensions of R m -valued 1-Lipschitz maps from a subset A of R n to a 1-Lipschitz maps on R n . We consider several cases when there exists a 1-Lipschitz extension with preserved uniform distance to a given 1-Lipschitz map. We prove that if m > 1, then a given map is 1-Lipschitz and affine if and only if such a distance preserving extension exists for any 1-Lipschitz map defined on any subset of R n . This shows a striking difference from the case m = 1, where any 1-Lipschitz function has such a property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map v is when the difference between the two maps takes values in a fixed one-dimensional subspace of R m and the set A is geodesically convex with respect to a Riemannian pseudo-metric associated with v.