Transport segments of a cyclically monotone set do not cross

Let E be a strictly convex real normed space and 1 < p. For the cost c (x, y) = d(x, y) ^ p, two pairs (x, x') and (y, y') of a c-cyclically monotone set cannot be moved along their segments so as to meet at an intermediate time: if (1 - t) • x + t • x' = (1 - t) • y + t • y' for some 0 < t < 1, then x = y and x' = y'. Equivalently, for every 0 < t < 1 the interpolation (x, x') ↦ (1 - t) • x + t • x' is injective on a c-cyclically monotone set.

Applied to the support of an optimal coupling, this says that in the displacement interpolation of an optimal plan no two particles occupy the same position at the same intermediate time. In Eulerian terms, the velocity of the interpolation is then the displacement itself, with no averaging over particles.

Proof

Suppose the two segments meet at the point z at time t, and write a = d(x, x') and b = d(y, y'). Going through z gives d(x, y') ≤ t a + (1 - t) b and d(y, x') ≤ t b + (1 - t) a, so by convexity of s ↦ s ^ p the swapped cost is at most a ^ p + b ^ p, which by cyclical monotonicity is at most the swapped cost. Strict convexity of s ↦ s ^ p forces a = b, and then d(x, y') = d(x, z) + d(z, y'). In a strictly convex space this equality case of the triangle inequality makes x' - x and y' - y point in the same direction; having the same length, they are equal, and the two segments coincide.

Main statements

• TauCeti.eq_and_eq_of_smul_add_smul_eq: two pairs satisfying the two-point swap inequality for the cost ‖· - ·‖ ^ p, 1 < p, whose segments meet at an intermediate time are equal.
• TauCeti.IsCyclicallyMonotone.injOn_smul_add_smul: for 1 < p and 0 < t < 1, the interpolation at time t is injective on every cyclically monotone set of the cost d ^ p.

References

• C. Villani, Optimal Transport: Old and New, Grundlehren 338, Springer 2009, Chapter 8, for the absence of crossings in displacement interpolation.
• L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, 2nd ed. 2008, Lemma 7.2.1.

theorem TauCeti.eq_and_eq_of_smul_add_smul_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {p t : ℝ} (hp : 1 < p) (ht : t ∈ Set.Ioo 0 1) {x x' y y' : E} (h : ‖x - x'‖ ^ p + ‖y - y'‖ ^ p ≤ ‖x - y'‖ ^ p + ‖y - x'‖ ^ p) (he : (1 - t) • x + t • x' = (1 - t) • y + t • y') : x = y ∧ x' = y'

Two-point no-crossing. In a strictly convex real normed space, let 1 < p and 0 < t < 1. If the pairs (x, x') and (y, y') satisfy the two-point swap inequality ‖x - x'‖ ^ p + ‖y - y'‖ ^ p ≤ ‖x - y'‖ ^ p + ‖y - x'‖ ^ p for the cost ‖· - ·‖ ^ p and their segments meet at time t, i.e. (1 - t) • x + t • x' = (1 - t) • y + t • y', then x = y and x' = y'.

theorem TauCeti.IsCyclicallyMonotone.injOn_smul_add_smul {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {p : ℝ} (hp : 1 < p) {S : Set (E × E)} (hS : IsCyclicallyMonotone (fun (z : E × E) => edist z.1 z.2 ^ p) S) {t : ℝ} (ht : t ∈ Set.Ioo 0 1) : Set.InjOn (fun (z : E × E) => (1 - t) • z.1 + t • z.2) S

Transport segments do not cross. In a strictly convex real normed space E, let 1 < p and let S be cyclically monotone for the cost d(x, y) ^ p. For every 0 < t < 1, the interpolation (x, x') ↦ (1 - t) • x + t • x' is injective on S: two pairs of S whose segments meet at time t are equal.