Suppose (M,d) is a metric space and $N\subset M$ is finite, $\varepsilon >0$. Also let there exist $u, v \in M$ and $u\neq v$ such that $$ (1-\varepsilon)(d(x,y)+d(u,v)) \leqslant d(x, u) +d(y, v) $$ $\forall x, y \in N$
Now if we consider an $1+\varepsilon$ -Lipschitz map $f:N\bigcup \{u, v\} \rightarrow \mathbb{R}$, then does it always admit an $1+\varepsilon$ -Lipschitz extension to the whole space M?