Continuity of extensions of Lipschitz maps and of monotone maps

Let $X$ be a subset of a Hilbert space. We prove that if $v\colon X\to \mathbb{R}^m$ is such that \begin{equation*} \Big\lVert v(x)-\sum_{i=1}^m t_iv(x_i)\Big\rVert\leq \Big\lVert x-\sum_{i=1}^m t_ix_i\Big\rVert \end{equation*} for all $x,x_1,\dotsc,x_m\in\mathbb{R}^m$ and all non-negative $t_1,\dotsc,t_m$ that add up to one, then for any $1$-Lipschitz $u\colon A\to\mathbb{R}^m$, with $A\subset X$, there exists a $1$-Lipschitz extension $\tilde{u}\colon X\to\mathbb{R}^m$ of $u$ such that the uniform distance on $X$ between $\tilde{u}$ and $v$ is the same as the uniform distance on $A$ between $u$ and $v$. Moreover, if either $m\in \{1,2,3\}$ or $X$ is convex, we prove the converse: we show that a map $v\colon X\to\mathbb{R}^m$ that allows for a $1$-Lipschitz, uniform distance preserving extension of any $1$-Lipschitz map on a subset of $X$ also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.