Results on multi-marginal transport, supporting potentials, local branches and Borel transport, with their assumptions

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.

• Supporting potentials — Proposition 2.1. If c >= u(x1)+u(x2)+u(x3) on U^3 with mu(U) = 1, u bounded, and some coupling is carried by the equality set, then the Kantorovich value is 3 times the integral of u, and every optimal coupling is carried by the equality set.
• Mass imbalance (OpenAI, after Gerolin, Kausamo and Rajala) — Lemma 3.1. For mu = nu/3 + (1/6) sum of four branch images with disjoint supports, no measure-preserving Borel map selects a branch at almost every central point.
• No Monge optimizer — Corollary 4.1. A supporting potential whose contact set offers only the four branches above each central point excludes every Monge optimizer.
• Smooth implicit and inverse functions — Lemma 1.1. Implicit functions and local inverses of smooth maps are smooth.
• Jets of the Riesz cost — Lemma 1.2. Hessian of |b|^-s with eigenvalues s(s+1)|b|^(-s-2) and -s|b|^(-s-2); at (0, a_k, -a_k) the cost is 2 + 2^-s, the x-gradient vanishes and the z-gradient is s(1 + 2^(-s-1)) a_k.
• A local branch (OpenAI) — Lemma 2.1. Near each triple (0, a_k, -a_k) there are diffeomorphisms X_k, Z_k and a smooth w_k with c >= u_0(x) + w_k(y) + v_k(z), equality exactly at x = X_k(y), z = Z_k(y).
• A potential on five balls (OpenAI) — Proposition 3.1. A bounded continuous potential on five small balls supports the Riesz cost with contact set exactly the permutations of the two branch families.
• The common ball — Lemma 3.2. On a ball B around 0 the maps Y_k = X_k^-1 and W_k = Z_k Y_k are diffeomorphisms onto disjoint open sets.
• The density — Lemma 4.1. mu = nu/3 + (1/6) sum of the four images of nu has a smooth compactly supported density with smooth square root.
• No Monge optimizer for Riesz costs (OpenAI) — Theorem 4.2. For every d >= 2 and s > 0, in particular for the Coulomb cost on R^3, the three-marginal minimum for mu is finite and attained, but every pair of measure-preserving Borel maps costs strictly more.
• Uniform variables — Lemma 1.1. If R has a continuous distribution function F, then F(R) is uniform on [0,1].
• Quantiles — Lemma 1.2. For a continuous distribution function, the rational quantile q satisfies q(u) <= t iff u <= F(t), so q(U) has distribution function F.
• Rosenblatt's transform — Lemma 2.1. Successive conditional distribution functions send a density on R^d to the uniform measure on the cube.
• Borel transport — Corollary 2.3. Any two Borel probability measures with densities on R^d are related by a Borel map Q with Q#alpha = beta.
• Splitting into prescribed masses — Lemma 3.1. A Borel set is a disjoint union of slabs of prescribed masses for a measure with a density.
• Monge plans approach the minimum (OpenAI) — Theorem 4.1. For the density of the second lesson, measure-preserving Borel maps come within every epsilon > 0 of the Kantorovich minimum without attaining it.
• Nonprincipal ultrafilters — Lemma 1.2. Zorn's lemma gives an ultrafilter on N that contains every set with finite complement.
• The hyperreal field — Lemma 1.4. The ultrapower of R by a nonprincipal ultrafilter is an ordered field containing R as an ordered subfield.
• Standard part — Lemma 1.6. Every finite hyperreal number is infinitely close to exactly one real number; the standard part is monotone.
• Limits by infinitesimals — Lemma 1.7. A real sequence converges to L if and only if its value at every infinite hypernatural number is infinitely close to L.
• Uniqueness of standard parts — Lemma 2.2. An internal Monge pair has at most one standard part: Borel probability measures on (R^d)^3 that agree on bounded continuous functions are equal.
• Cell plans — Lemma 3.1. For every delta > 0 the cell construction gives measure-preserving maps with energy in (E, E + 6 s delta eta^(-s-1)] whose plan gives every product of three cells the same mass as pi_0.
• The theorem in infinitesimal form — Theorem 3.2. Every internal Monge pair has energy above E; at an infinite hypernatural M the cell pair exceeds E by at most 6 s eta^(-s-1)/M, and its standard part is pi_0, which is not a Monge plan.
• Internal mass imbalance — Proposition 3.3. An internal Monge pair agrees with a branch only on an internal set of nu-measure exactly zero.
• Branch following — Proposition 4.1. The internal map is infinitely close to some branch at almost every central point and follows each branch on a quarter of the central mass, weighted by any continuous function.
• Branch switching at infinitesimal scale — Corollary 4.2. Inside every open set charged by nu, each branch is followed on a part of mass that is not infinitesimal.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.