Show that there exist three laws α, β, and γ on R2 such that there is no law P on R6 with coordinates x, y, and z in R2 for which x, y, and z have laws α, β, and γ , respectively, and E |x - y|= W (α, β), E |y - z|= W (β, γ ), and E |x - z|= W (α, γ ). Hint: Let a, b, and c be the vertices of an equilateral triangle of side 1. Let α := (δa + δb )/2,β = (δa + δc )/2, and γ = (δb + δc )/2. (So the simultaneous attainment of Monge-Wasserstein distances by random variables for all laws, as in R1 in Problem 2, no longer is possible in R2.)