Results on Wulff bodies, the reduction of the logarithmic Brunn-Minkowski inequality, and moment pairs

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.

• Support functions of convex bodies — Lemma 1.1. The support function of a convex body is Lipschitz; the body is the set of the x with <x,u> <= h_K(u) for all unit vectors u; for K = -K it is even and bounded below by a positive number; it is additive for Minkowski combinations.
• Wulff bodies — Lemma 1.2. The Wulff body W[f] of a positive continuous function on the sphere is a convex body between the balls of radius min f and max f; it is monotone and homogeneous in f; W[h_K] = K; W[h_K^(1-l) h_L^l] lies in (1-l)K + lL for symmetric K, L.
• The logarithmic Brunn-Minkowski inequality follows from the variance bound — Theorem 2.2. If Var(sum b_i psi_i) <= sum b_i^2 E psi_i holds for all densities proportional to exp(-eps|x|^2/2 - sum psi_i), psi_i = (<p_i,x>/h_i)^q, in dimension n >= 2, then |W[h_K^(1-l) h_L^l]| >= |K|^(1-l) |L|^l for all origin-symmetric convex bodies in R^n; in dimension 1 equality holds.
• Second derivative of the logarithm of the integral of a soft polytope — Lemma 3.1. For widths h_i(t) = (h_i^0)^(1-t) (h_i^1)^t: (log Z)'' = Var(sum b_i psi_i) - sum b_i^2 E psi_i with b_i = -q log(h_i^1/h_i^0).
• The additive inequality for 0 < p < 1 from the logarithmic one — Corollary 6.1. If the logarithmic Brunn-Minkowski inequality holds in dimension n, then |W[((1-l) h_K^p + l h_L^p)^(1/p)]|^(p/n) >= (1-l)|K|^(p/n) + l|L|^(p/n) for origin-symmetric convex bodies and 0 < p < 1.
• The moment equation of a moment pair — Lemma 1.1. If grad phi is a diffeomorphism of R^n with positive definite Hessian that carries e^-phi dz to Z^-1 e^-V dx, then log det D^2 phi = V(grad phi) + log Z - phi.
• Adjoints in moment coordinates — Lemma 2.1. For a moment pair with x = grad phi(z), tau = D^2 phi: the adjoint of d/dz_k in L^2(e^-phi) is x_k - d/dz_k; sum_k (x_k - d/dz_k) P_ak = (delta_mu (P tau))_a; and delta_mu tau = x, that is, E x_j g = E sum_k tau_jk g_{x_k}.
• The operator A of a moment pair — Proposition 3.1. A f = delta_mu (tau grad f) = x . grad f - tau : D^2 f satisfies E g A f = E grad g . tau grad f, A 1 = 0, A x_j = x_j, and A(fg) = f A g + g A f - 2 grad f . tau grad g.
• Test fields of a moment pair and the Hessian identity — Proposition 4.1. For f with compact support, B = tau D^2 f tau, W = delta_mu B, u = delta_mu W: E u F = E W . grad F = E B : D^2 F for smooth F; W = tau grad (A f - f); u = A(A-1) f; and E (A f)(A f - f) = E |tau^(1/2) D^2 f tau^(1/2)|^2.
• The tensor estimate in moment coordinates — Theorem 1.1. For a moment pair, f with compact support, B = tau D^2 f tau, W = delta_mu B, u = delta_mu W, H = D^2 V, M = tau^-1: E tr((D_x W)^2) >= E tr(M B H B) and E u^2 >= E tr(M B H B) + E W^T H W.
• A weighted identity for the divergence of a vector field — Lemma 2.1. For a smooth vector field W with compact support and u = delta_mu W: E u^2 = E W^T (D^2 V) W + E tr((D W)^2).
• The moment equation differentiated twice — Lemma 3.2. For a moment pair with C = D^3 phi: (tau H tau - tau)_ij = (d_d - x_d)(M_dk C_ijk) - tr(M C_i M C_j).
• A sum of squares for tensors with three indices — Lemma 6.1. For S fully symmetric and J symmetric in its last two indices: 2|S|^2 + 4<S,J> + |J|^2 + <J,J^(12)> = 2|S + Sym J|^2 + |J - J^(12)|^2 / 6.
• Density of the even tests of a moment pair — Theorem 1.1. Assuming interior elliptic regularity (ER): for an even moment pair with tau <= Lambda, the functions A(A-1)f with f even, smooth and compactly supported are dense in the even functions of L^2(mu) with mean zero.
• Square-integrable solutions of A(A-1)g = 0 are affine — Proposition 5.1. For a moment pair with tau <= Lambda: a smooth g in L^2(mu) with A(A-1)g = 0 is affine.
• The dual variance bound on the test functions — Proposition 3.1. Assuming (M): for u = A(A-1)f with f smooth and compactly supported, sum_i (E u psi_i)^2 / E psi_i <= E u^2.
• The variance bound for soft polytopes — Theorem 4.1. Assuming (M) and (ER): for mu proportional to exp(-eps|x|^2/2 - sum psi_i), psi_i = (<p_i,x>/h_i)^q with q even, in dimension n >= 2: Var(sum b_i psi_i) <= sum b_i^2 E psi_i.
• The logarithmic Brunn-Minkowski inequality — Theorem 5.1. Assuming (M) and (ER): |W[h_K^(1-l) h_L^l]| >= |K|^(1-l) |L|^l for all origin-symmetric convex bodies K, L in R^n, every n >= 1 and 0 <= l <= 1; and the additive inequality for 0 < p < 1.
• The variance bound for Gaussian soft polytopes — Proposition 6.1. For q = 2 and every dimension, without hypotheses: Var(sum b_i psi_i) <= sum b_i^2 E psi_i, where psi_i = <alpha_i, x>^2 and mu is the Gaussian measure with density proportional to exp(-eps|x|^2/2 - sum psi_i).

Reading order

Five lessons with the whole argument of OpenAI's proof of the logarithmic Brunn-Minkowski inequality: Wulff bodies and the reduction to a variance bound; moment pairs, their adjoint identities and the operator A; the tensor estimate; the density of the even test functions; the variance bound and the inequality. Two statements from the literature are named hypotheses and are not proved in the course: (M), the existence of moment coordinates with a bounded Hessian, and (ER), interior regularity for an elliptic operator of order 4. The inequality is proved from these two.