Kirszbraun extensions preserving uniform distance in Hilbert spaces

Abstract

Let $X$ be a subset of a real Hilbert space and let $v\colon X\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\subset X$, $ρ\geq0$, and $u\colon A\to Y$ is $1$-Lipschitz with $\left\lVert u(x)-v(x)\right\lVert\leqρ$ for $x\in A$, there is a $1$-Lipschitz extension $\widetilde u\colon X\to Y$ with $\left\lVert \widetilde u(x)-v(x)\right \lVert\leqρ$ for $x\in X$; and for every $1\leq k\leq\dim Y$, $$ \left\lVert v(x_0)-\sum_{i=1}^k t_i v(x_i)\right\lVert \leq \left\lVert x_0-\sum_{i=1}^k t_i x_i \right \lVert$$ whenever $x_0,\ldots,x_k\in X$, $t_1,\ldots,t_k\geq0$, and $\sum_{i=1}^k t_i=1$. Previous necessity results required $\dim Y\leq3$ 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 $\dim\operatorname{Aff}X+1\leq\dim Y$, we also obtain a lifting theorem for convex Lipschitz functions and transfer convex Poincaré inequalities without increasing the constant.

Disclosure

“16 KRZYSZTOF J. CIOSMAK Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the author used OpenAI Codex to assist in developing and checking mathematical arguments, organising the manuscript and”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T18:58:36.279052Z

Structural counts