A metric space <i>(V,d)</i> will be called distance regular if for every distances <i>a&gt;0, b, c</i> a nonnegative integer <i>p(a,b,c)</i> is defined, so that whenever <i>d(B,C)=a</i>, there are precisely <i>p(a,b,c)</i> points *A* such that *d(A,B)=c, d(A,C)=b*. 

The Euclidean plane is an example: *p(a,b,c)=0,1,* or *2* when the triangle inequality for *a,b,c*, correspondingly, fails, turns into equality, or is strict.

If we also require that *p(a,b,c)&gt;0* whenever the triangle inequality does not fail, then I conjecture that this is the only possibility for the parameters <i>p(a,b,c)</i>. 

Has anybody formulated/proved/refuted this conjecture before? It looks very natural.