Blend by multiplying?

0:00Blending shapes

Take two convex shapes, K and L. The classical Brunn–Minkowski inequality says that blending them, by averaging every point of K with every point of L using fixed weights, gives a shape whose volume is at least the weighted geometric mean of their volumes. Describe a convex shape around the origin by its support function: how far it reaches in each direction. The classical blend averages support functions. The logarithmic version multiplies them instead: take the geometric mean of the two reaches in every direction, and keep the points that stay within that reach in every direction. That shape sits inside the classical blend, so a volume bound for it is stronger. For a square and the same square turned 45 degrees, both of area 4, the classical midpoint, a regular octagon, has area 4.83, and the logarithmic one, slightly smaller, 4.69, still above 4. Böröczky, Lutwak, Yang and Zhang conjectured in 2012 that for shapes symmetric about the origin, the logarithmic blend always has volume at least the geometric mean, and they proved it in the plane. Symmetry matters: shift a square of side 2 sideways by 0.6, keeping the origin inside, and the logarithmic midpoint of the square and its shifted copy has area 3.79, below 4. Equality holds for dilates of one shape, and for parallelograms with parallel sides. In our computation over 300 random pairs of symmetric polygons, the smallest ratio of the logarithmic blend’s area to the geometric mean was 1.025, and over 80 pairs of random symmetric polyhedra, 1.18. Many special cases were known. Saroglou proved it for bodies symmetric under every sign change of the same coordinates, Böröczky and Kalantzopoulos for pairs sharing enough reflection symmetries, and Rotem for unit balls of complex norms. A second line studied small perturbations: Colesanti, Livshyts and Marsiglietti near the ball, Kolesnikov and Milman through an eigenvalue problem, and van Handel for zonoids, while Putterman showed that local versions imply global ones. Stancu announced a proof in 2018, with details of a flow argument deferred. Between the logarithmic and the classical inequality sit the Lᵖ versions, for p between 0 and 1. And nearby is the (B)-conjecture: for an even log-concave measure, the measure of eᵗ times a symmetric body should be log-concave in t. Cordero-Erausquin, Fradelizi and Maurey proved it for the Gaussian measure in 2004.

2:46The claim

Result 091 is one OpenAI preprint, 20 pages, dated September 23. It claims the logarithmic Brunn–Minkowski inequality for every pair of origin-symmetric convex bodies, in every dimension, with no smoothness assumption. As corollaries, it claims the symmetric Lᵖ inequality for every p between 0 and 1, and, through a transfer theorem of Saroglou, the same logarithmic inequality for every even log-concave measure, and with it the (B)-conjecture. The claims are for symmetric bodies only; without symmetry it can fail, as the shifted square shows. The paper does not determine the equality cases in higher dimensions, and the (B)-conjecture it proves concerns dilations by a single scalar. OpenAI also published a Lean formalization of the main theorem, a single file of about 24,500 lines, matched against a separate statement. The Lᵖ and measure corollaries are not formalized. We have not run the check.

3:52How the proof goes

The proof begins with slabs. Finitely many symmetric slabs, each between two parallel hyperplanes, cut out a symmetric polytope. Replace the polytope’s indicator by a smooth weight: e to the minus a small Gaussian term, minus a sum of even powers of the slab coordinates. As the power grows and the Gaussian term fades, the integral of this weight becomes the polytope’s volume. So it is enough that the logarithm of the integral is concave when each slab width changes geometrically. Its second derivative is a variance minus an average, so everything reduces to one variance bound: the variance of a weighted sum of the power terms is at most the sum of the squared weights times their averages. In our tests on 120 random planar cases, the largest ratio was 0.9975, never above one, and in one dimension, where the volume inequality is an equality, it creeps toward one as the Gaussian term vanishes. To prove the variance bound, the paper changes coordinates. Following the moment-measure theory of Cordero-Erausquin and Klartag, and using a theorem of Berman and Berndtsson and a Hessian estimate of Klartag, it writes the probability as the image of a measure e^(−φ) under the gradient of a convex function φ, whose Hessian is positive and bounded above. In these coordinates, a differential operator produces test functions that are dense among centered even functions. The only obstructions are affine functions: centering removes constants, and evenness removes linear ones. This is the key place symmetry enters. Then a tensor calculation leaves a remainder that is a sum of explicit squares, and two Cauchy–Schwarz inequalities give the bound. The rest is limits and known transfers. Letting the power grow and the Gaussian term vanish gives the inequality for polytopes, and adding slab directions densely gives every symmetric body. The Lᵖ inequality follows by the arithmetic–geometric mean inequality after normalizing volumes. For measures, Saroglou showed that the volume inequality transfers to every even log-concave density in the same dimension, and a reduction handles measures living on a subspace; applying it to two dilates of one body gives the (B)-conjecture. For the density e^(−|x₁| − |x₂|), and a tilted square, our computed log measure of eᵗ times the square is concave in t, as the theory predicts.

6:27Where it stands

The preprint is not peer reviewed and has not been revised. OpenAI’s Lean formalization covers the main theorem; we have not run it. The Lᵖ corollary is a short argument in the paper, while the measure results rest on Saroglou’s transfer theorem and Borell’s description of log-concave measures. The paper acknowledges Stancu’s earlier announcement. It is not among the papers OSVerify reviewed, and we have seen no expert review. If it holds, the logarithmic Brunn–Minkowski conjecture is settled for symmetric bodies in every dimension, together with the symmetric Lᵖ inequalities for p between 0 and 1, and the (B)-conjecture for even log-concave measures. The equality cases in general are not settled here.