= Interpolating metric space
A <metric space> $(X,d)$ is interpolating if, for every $x,y\in X$ and every pair of <nonnegative real numbers> $r,s$ with $d(x,y)=r+s$, there is $z\in X$ with $d(x,z)=r$ and $d(z,y)=s$. Thus each distance can be split at any prescribed fraction. The nonnegativity requirement is essential.
A <convex set> in a <normed vector space> is interpolating: when $r+s>0$, take $z=x+\frac{r}{r+s}(y-x)$; when $r+s=0$, take $z=x=y$. A two-point <metric space> with the <discrete metric> is not interpolating, because a distance of one has no midpoint at distance $1/2$ from both endpoints. Interpolation makes the <metric exponential candidate> satisfy the <triangle inequality>, so every bounded interpolating <metric space> is an <exponentiable object> of the <category of bounded metric spaces and non-expansive maps>.
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