A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) 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)>0 whenever the triangle inequality does not fail, then I conjecture that this is the only possibility for the parameters p(a,b,c). That is, there may be many non-isomorphic examples, but the parameters will be the same for all of them. (Thanks to Heather for this clarification.)

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