Kirszbraun extensions preserving uniform distance in Hilbert spaces

Let $X$ be a subset of a real Hilbert space and let $v\text{\hspace{0.17em}⁣}:X\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\subset X$, $\rho \ge 0$, and $u\text{\hspace{0.17em}⁣}:A\to Y$ is $1$-Lipschitz with $\parallel u(x)-v(x)\parallel \le \rho$ for $x\in A$, there is a $1$-Lipschitz extension $\tilde{u}\text{\hspace{0.17em}⁣}:X\to Y$ with $\parallel \tilde{u}(x)-v(x)\parallel \le \rho$ for $x\in X$; and for every $1\le k\le \dim Y$, $$\parallel v(x_0)-\sum _{i=1}^k{t}_{i}v(x_i)\parallel \le \parallel x_0-\sum _{i=1}^k{t}_i{x}_i\parallel$$ whenever $x_0,\dots ,x_k\in X$, $t_1,\dots ,t_k\ge 0$, and ${\sum }_{i=1}^k{t}_i=1$. Previous necessity results required $\dim Y\le 3$ 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 $\dim \mathrm{Aff}X+1\le \dim Y$, we also obtain a lifting theorem for convex Lipschitz functions and transfer convex Poincaré inequalities without increasing the constant.