Consistent clustering

Suppose we are in the clustering problem set up. A clustering algorithm $C$ is considered consistent if for distance $d$ on $T$ we get clustering $C(d) = f$ then any distance $d'$ such that

• $d'(a,b) \leq d(a,b)$ for any $a,b \in T$ where $f(a) = f(b)$ , and
• $d'(a,b) \geq d(a,b)$ for any $a,b \in T$ where $f(a) \not = f(b)$

then $C(d') = f$.