Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with distcombinatorial(vi, vj) = shortest path of edges between vi and vj.
That means: for each vi1, vj1, vi2, vj2:
disteuclidean(vi1, vj1) = disteuclidean(vi2, vj2)
distcombinatorial(vi1, vj1) = distcombinatorial(vi2, vj2)
Question 1: Is property X already named? What's its common name?
Question 2: Which polytopes have property X? The regular polytopes seem to have it, but are there more?