[math-fun] Metric space question

Consider the plane endowed with the L^p metric: d(u,v) := ((u_2-u_1)^p + (v_2-v_1)^p)^(1/p). This is in fact a metric just for 1 <= p <= oo, so assume 1 <= p <= oo. Call this metric space M_p. QUESTION: For which p is M_p isometric to a subset X of some Euclidean space R^N (necessarily homeomorphic to a plane) endowed with the "induced metric" ? ( Note: The induced metric d_I on a path-connected subset of R^n is defined as d_I(w,z) := inf {length(alpha) | alpha is a path in X connecting w and z} ) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele