Results on optimal maps, Laguerre cells, cell moments and gradient interpolation

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Null boundaries and sections of convex bodies — Lemma 1.1. A compact convex set with nonempty interior has a null boundary; half-open segments from interior points stay interior; an affine section through the interior has relative interior equal to the section of the interior and a null relative boundary.
• Optimality of finite affine maxima — Lemma 2.1. The gradient of a finite maximum of affine functions is the unique quadratic optimal map from the uniform measure on a convex body to its image measure, and every optimal plan lives on its graph.
• One-third stability of Brenier maps — Theorem 3.1. For the uniform measure on a convex body and targets in a fixed compact set, optimal maps exist, are unique, and satisfy ||T_mu - T_nu||_{L^2} <= C W_2(mu,nu)^(1/3) with one constant; the exponent cannot be increased.
• Sharpness on the cube — Proposition 4.1. On the cube with three-atom targets, tilting the central affine function keeps its mass but moves a set of mass b/2 between slopes at distance one, so the map distance is of order a^(1/2) while W_2 is of order a^(3/2).
• Derivatives of Laguerre cells — Proposition 1.1. Masses and first moments of Laguerre cells with positive masses are continuously differentiable in the intercepts and sites, with face-integral formulas; the face graph is connected and the mass Laplacian has kernel the constants.
• Intercepts for prescribed masses — Proposition 2.1. For positive weights and distinct sites there are unique normalized intercepts whose cells have the prescribed masses, and they depend continuously differentiably on the sites.
• A midpoint coupling — Lemma 1.1. Two uniform laws on compact convex sets U and V can be coupled so that the midpoint has density at most 1/sqrt(|U||V|), by Knothe's triangular rearrangement.
• The moment estimate — Proposition 2.1. With fixed sites, the weighted squared change of the cell moments is at most 162 R^2 times the weighted squared change of the cell masses, uniformly in the number and sizes of the cells.
• Potentials of finite targets — Proposition 1.1. Two finite targets in B(0,L) have affine potentials whose gradients are their optimal maps and whose L^2 distance is at most (1 + sqrt 162) R W_2.
• Gradients of convex functions — Lemma 1.1. A convex Lipschitz function has partial derivatives almost everywhere, is differentiable wherever all of them exist, and has a unique supporting slope there.
• Gradient interpolation — Lemma 1.2. For convex L-Lipschitz functions on a convex body, the squared L^2 distance of the gradients is at most 12d h^-2 times that of the functions plus 28 L^2 P_K h/|K|.
• Stability for finite targets — Corollary 1.3. For finite targets, ||T_mu - T_nu||_{L^2} <= C_* W_2^(1/3) with an explicit constant depending only on the body and the radius of the targets.
• Existence and uniqueness of optimal maps — Proposition 2.1. Every target in a compact set has a unique quadratic optimal map from the uniform measure on a convex body, obtained as the limit of the maps of finite discretizations.
• The theorem for all targets — Theorem 2.2. The one-third stability estimate holds for all Borel probability measures on the compact set, with the same constant.

Reading order

Five lessons, to be read in order. The theorem is proved in full; measure theory, the implicit function theorem and the Arzela-Ascoli theorem are taken from the core courses Measure and Integration and Real Analysis II and cited where they are used.