Assume we have $n$ points $p\_0\ldots p\_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p\_0) = 0$, $f(p\_1) = d(p\_0,p\_1)$ and $d(p\_i,p\_j) \geq |f(p\_i) - f(p\_j)|$ for all $i,j$?


If the $p\_i$ are points in a vector space with inner product, $f$ could be just the orthogonal projection onto the ray of direction $p\_j - p\_i$, but for general metrics, I do not know whether such a function exists.



Generalised "projection" of a metric space
Take $f(x)=d(x,p\_0)/2-d(x,p\_1)/2+d(p\_0,p\_1)/2$.


