  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
$v$.
  Morever, 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.
