Kirszbraun extensions preserving uniform distance in Hilbert spaces

2026/07/20 by Krzysztof J. Ciosmak
#math.FA #math.MG

paper · pdf

Abstract

Let X be a subset of a real Hilbert space and let v\colon X→ Y, where Y is a real Hilbert space. We prove that the following conditions are equivalent: whenever A⊂ X, ρ≥0, and u\colon A→ Y is 1-Lipschitz with ‖ u(x)-v(x)‖≤ρ for x∈ A, there is a 1-Lipschitz extension \widetilde u\colon X→ Y with ‖ \widetilde u(x)-v(x) ‖≤ρ for x∈ X; and for every 1≤ k≤dim Y, ‖ v(x₀)-∑_i=1^k t_i v(x_i)‖ ≤ ‖ x₀-∑_i=1^k t_i x_i ‖ whenever x₀,…,x_k∈ X, t₁,…,t_k≥0, and ∑_i=1^k t_i=1. Previous necessity results required dim Y≤3 or convexity of X. For finite-dimensional targets, an application gives an exact data processing characterisation for a finite branching hierarchy connecting Wasserstein and barycentric weak transport. If Y is infinite-dimensional or \dimAffX+1≤dim Y, we also obtain a lifting theorem for convex Lipschitz functions and transfer convex Poincaré inequalities without increasing the constant.

Related

• Continuity of extensions of Lipschitz maps and of monotone maps
• Kirszbraun's theorem via an explicit formula
• Continuity of extensions of Lipschitz maps
• Extension operators and nonlinear structure of Banach spaces
• Linear extension operators between spaces of Lipschitz maps and optimal transport