Proof. See pp. 44 − 46 of Villani (2003). In the following discussion, I focus on the lower bound on the DTE since the procedure to obtain the upper bound is similar. Remark 2 In the proof of Lemma 3, Villani (2003) showed that at the optimum, A = { x ∈$_{R}$| ϕ ( x ) ≥ s } for some s ∈ [0 , 1] . Since the function ϕ is continuous, if ϕ is nondecreasing then A = [ a, ∞ ) for some a ∈ [ −∞ , ∞ ] where A = φ if a = ∞ . In contrast, if ϕ is nonincreasing, then A = ( −∞ , a ] where A = φ if a = −∞ Remember that for any ( y$_{0}$, y$_{1}$ ) in the support of the optimal joint distribution, ϕ and ψ satisfy Pick ( y ′ $_{0}$, y ′ $_{1}$) and ( y ′′ $_{0}$, y ′′ $_{1}$) with y ′′ 0 > y ′ 0 in the support of the optimal joint distribution. Then, The inequality in the second line of (8) is obvious from (7) and the inequality in the third line of (8) holds because 1 { y$_{1}$ − y$_{0}$ < δ } is nondecreasing in y$_{0}$ . Since ϕ is nondecreasing on the set { y$_{0}$ ∈ Y$_{0}$|∃ y$_{1}$ ∈ Y$_{1}$ s.t. ( y$_{0}$, y$_{1}$ ) ∈ Supp ( π ) } by Remark 2 A can be written as [ a, ∞ ) for some a ∈ [ −∞ , ∞ ] . 15 ϕ ( y ′ $_{0}$) = 1 { y ′ 1 − y ′ 0 < δ } − ψ ( y ′ $_{1}$) (8) ≤ 1 { y ′′ 1 − y ′ 0 < δ } − ψ ( y ′′ $_{1}$) ≤ 1 { y ′′ 1 − y ′′ 0 < δ } − ψ ( y ′′ $_{1}$) = ϕ ( y ′′ $_{0}$) . ϕ ( y$_{0}$ ) = inf y$_{1}$ ∈$_{R}$ { 1 { y$_{1}$ − y$_{0}$ < δ } − ψ ( y$_{1}$ ) } . (7) Figure 6: A D for A = [ a, ∞ )