Definition: A polytope has **property X** iff there is a function f:N^{+} → R^{+} such that for each pair of vertices v_{i}, v_{j} the following holds:

dist_{euclidean}(v_{i}, v_{j}) = f(dist_{combinatorial}(v_{i}, v_{j}))

with dist_{combinatorial}(v_{i}, v_{j}) = shortest path of edges between v_{i} and v_{j}.

That means: for each v_{i1}, v_{j1}, v_{i2}, v_{j2}:

dist_{euclidean}(v_{i1}, v_{j1}) = dist_{euclidean}(v_{i2}, v_{j2})

iff

dist_{combinatorial}(v_{i1}, v_{j1}) = dist_{combinatorial}(v_{i2}, v_{j2})

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?