Kirszbraun's Theorem and Metric Spaces of Bounded Curvature

Abstract

We generalize Kirszbraun's extension theorem for Lipschitz maps between (subsets of) euclidean spaces to metric spaces with upper or lower curvature bounds in the sense of A.D. Alexandrov. As a byproduct we develop new tools in the theory of tangent cones of these spaces and obtain new characterization results which may be of independent interest.

Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

“…These results were previously proved in [36] via the method of random partitions (Lipschitz extension for spaces of bounded Nagata dimension was previously treated in [34] and only later it was shown in [52] that they admit a padded random partition and therefore the corresponding extension results are a special case of [36]). It also follows that if (X X ) has nonnegative curvature in the sense of Aleksandrov and (Y Y ) is a Hadamard space then (X Y ) 1, a result that has been previously proved in [35], as a special case of an elegant generalization of the classical Kirszbraun extension theorem [31]. be extended to a Y -valued 1/2-Hölder mapping defined on all of X ; this statement was previously known when Y is a Hilbert space due to the work of Minty [47].…”

The fixed point property and unbounded sets in CAT(0) spaces

“…112] or [20,22]): Let G 1 (C) denote the union of all geodesics segments with endpoints in A. Notice that C is convex if and only if G 1 (C) = C. Recursively, for n ≥ 2 we set G n (C) = G 1 (G n−1 (C)).…”

Metric Currents and Geometry of Wasserstein Spaces

“…with convex metric. We refer to [23,24,25,28,29,30,33] for precise definitions and main properties. Here we focus our attention on Wasserstein spaces.…”

“…The following statement, adapted to the case of curvature 0 and ! 0 in the Alexandorov sense, is taken from [25].…”