Continuity of extensions of Lipschitz maps

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