Kirszbraun theorem

Every 1-Lipschitz map from a subset of one Hilbert space into another extends to a 1-Lipschitz map on the entire domain space. The extension preserves the original map’s Lipschitz bound.

Lipschitz map: A map between metric spaces whose output distances are bounded by a constant times input distances. The theorem extends maps whose Lipschitz constant is at most one.
Hilbert space: A complete inner-product space, with distance induced by its inner product. The theorem’s source and target spaces must have Hilbert geometry.
Mojżesz David Kirszbraun: A Polish mathematician who proved the extension theorem now bearing his name. His 1934 work established the result for Euclidean spaces.
Kirszbraun-Valentine theorem: The Hilbert-space formulation of Kirszbraun’s Lipschitz extension theorem, including infinite-dimensional spaces. This names the extension from Euclidean spaces to Hilbert spaces.
Kirszbraun's theorem for CAT(0) spaces: An extension theorem for 1-Lipschitz maps from subsets of certain nonpositively curved spaces. This is a distinct generalization whose validity depends on curvature assumptions.
Closed ball: The set of points at distance no greater than a specified radius from a center in a metric space. Candidate values for an added point lie in balls centered at existing image points.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Metric distance defines the Lipschitz condition being preserved.
Heinz König: A German mathematician known for work in analysis, topology, and set theory. König helped develop and present extensions of the theorem’s proof and scope.
Whitney extension problem: The problem of extending functions or prescribed derivatives from a subset while retaining specified smoothness. Kirszbraun gives a metric analogue of extending data while preserving a regularity bound.
Aronszajn-Panitchpakdi theorem: A characterization of hyperconvex metric spaces through extension of nonexpansive maps. It identifies a target-space property that ensures extensions from arbitrary metric domains.

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