Logarithmic and L_p Brunn–Minkowski inequalities and the B-conjecture

Blending two symmetric shapes the thrifty way never loses volume below their geometric mean.

1 paper September 23, 2026 Explained by AI

Two shapes can be blended by taking, in each direction, the geometric mean of how far each one reaches. The logarithmic Brunn–Minkowski conjecture says this blend never has less volume than the geometric mean of the two volumes. This paper proves it for all convex shapes symmetric about the origin, in every dimension (Lean proof listed). It also claims the Lp version for powers between 0 and 1. It further claims the scalar B-conjecture for even log-concave measures.

First step

2012

Böröczky, Lutwak, Yang and Zhang, conjecture posed and planar case proved

Significance

Major

It settles a well-studied conjecture in convex geometry, where earlier proofs needed extra symmetry or nearness to a ball.

Trust

Formal proof listed

Field

Convex and metric geometry

1 paper

The explanation

Take two boxes centred on the same point. One is 4 units wide and 1 unit tall. The other is 1 unit wide and 4 units tall. Each has area 4. Now blend them halfway. The usual blend averages each point of one box with each point of the other. Here it gives a square 2.5 units on a side, with area 6.25.

There is a thriftier blend. Describe each shape by how far it reaches in every direction. The wide box reaches 2 to the right and 0.5 upward. The tall box reaches 0.5 to the right and 2 upward. Instead of averaging the reaches, multiply them and take the square root. That is called the geometric mean. To the right it gives 1, and upward it also gives 1. The blend is the largest shape that stays within these limits in every direction. Here it is a square 2 units on a side. Its area is 4, exactly the geometric mean of the two areas, √(4 × 4).

Shapes need not line up so neatly. Take a square 2 units on a side, with area 4. Take a diamond whose corners sit 1 unit from the centre, with area 2. The usual halfway blend has area 3.5. The thrifty blend is the square with its four corners trimmed off. Its area is 8√2 − 8, about 3.31. The geometric mean of the two areas is √8, about 2.83. So the thrifty blend is smaller than the usual one, yet still above that mark.

The logarithmic Brunn–Minkowski conjecture says this always happens. Take any two convex shapes (no dents or holes) in any number of dimensions. Each must be origin-symmetric (unchanged when flipped through its centre). Then the thrifty blend never has less volume than the geometric mean of the two volumes. This also holds for uneven blends, such as 30 percent of one and 70 percent of the other. The thrifty blend always fits inside the usual one. So in the symmetric setting this strengthens the classical Brunn–Minkowski inequality, which makes the same promise for the usual blend.

Böröczky, Lutwak, Yang and Zhang posed the conjecture in 2012 and proved it in the plane. Saroglou proved it in 2015 for shapes unchanged by every flip of a coordinate sign, like the boxes above. In 2022, Böröczky and Kalantzopoulos extended this to shapes sharing n mirror symmetries whose mirrors meet only at the centre. Other work proved local versions, for shapes very close to a ball. In 2018, Stancu announced the full inequality in a short report, but the report deferred the details.

The main obstacle is this. The geometric mean of two reach profiles need not be the reach profile of any shapeSource 1. The blend must be built by cutting space with all those limits at once. The proof must work with that cut throughout. Earlier proofs leaned on extra symmetry or stayed near a ball. This paper approximates each shape by many flat stripsSource 2. It then softens the strips' hard edges into smooth fading weights. The question becomes one inequality about how much a random quantity can fluctuateSource 3.

Two further results follow. The first is the Lp version for every power p between 0 and 1Source 4. It blends reaches with a power mean instead of the geometric mean. The paper claims it follows from the logarithmic case by a known argument. The second replaces volume by an even log-concave measure. That is a way of weighting space that looks the same when flipped through the centre. Blending two sets the usual way never gives less weight than the geometric mean of their weights. A bell curve centred at the origin is one example. Spreading weight evenly over a symmetric convex shape, and none outside it, is another. Using Saroglou's 2016 transfer theorem, the paper claims the same inequality for these measuresSource 5. It also claims the B-conjecture in its scalar formSource 6. Scale a symmetric shape by a factor e^t. Then the logarithm of its measure is a concave function of t, meaning it bends downwardSource 7.

These results need the symmetry about the centre. The theorem says nothing about shapes that are lopsided around the origin. The B-conjecture part covers scaling one shape by a single factor. It does not cover stretching different directions by different amounts.

How certain is this? A computer proof checker called Lean has checked a formal version of the main theorem. It covers every dimension and all origin-symmetric convex shapes, with no smoothness assumption. The Lp corollary and the results for measures are outside that check. No specialist review has been published yet.

What it doesn't show. This does not cover convex shapes that are not symmetric about the origin.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. A Lean formalization of the main volume inequality is listed. The Lp and measure consequences are outside that check, and no specialist review has been published.

Words used

Convex shape: A shape with no dents or holes: the straight segment between any two of its points stays inside.
Origin-symmetric: Unchanged when every point x is replaced by its mirror point −x through the centre.
Geometric mean: For two positive numbers, the square root of their product; for uneven weights, a weighted version of that.
Brunn–Minkowski inequality: A classical lower bound on the volume of the usual blend of two shapes.
Lp blend: A blend that combines the reaches of two shapes by a power mean with power p instead of the geometric mean.
Log-concave measure: A way of weighting space in which the usual blend of two sets never weighs less than the geometric mean of their weights. Examples: a bell curve, or even weight spread over a convex shape.
B-conjecture: The claim that, for such a weighting, the log of a symmetric shape's weight is concave as the shape is scaled up exponentially.

The original paper

The logarithmic Brunn–Minkowski conjecture (PDF, opens in a new tab)
September 23, 2026 Formal proof listed

Lean notes

All quoted lines (7)

In general $h_K^{1-\lambda}h_L^\lambda$ need not itself be a support function.
The logarithmic Brunn–Minkowski conjecture Back to text
As $q$ tends to infinity, this weight becomes the Gaussian weight restricted to the polytope.
The logarithmic Brunn–Minkowski conjecture Back to text
The analytic task is therefore a variance bound for sums of even powers of linear forms (Proposition …).
The logarithmic Brunn–Minkowski conjecture Back to text
The logarithmic endpoint implies the full symmetric volume inequality in this intermediate range.
The logarithmic Brunn–Minkowski conjecture Back to text
Saroglou proved that the Lebesgue inequality in dimension $n$ implies the same Wulff inequality for every such density in that same dimension.
The logarithmic Brunn–Minkowski conjecture Back to text
The diagonal statement here is restricted to uniform measures on cubes; the consequence for general even log-concave measures concerns scalar dilations.
The logarithmic Brunn–Minkowski conjecture Back to text
Taking two scalar dilates of a single body then gives the $(B)$ -conjecture: for the measure $d\mu=e^{-V(x)}\,dx$ and every origin-symmetric convex body $K$ , the function $t\mapsto\mu(e^tK)$ is log-concave on $\R$ .
The logarithmic Brunn–Minkowski conjecture Back to text

Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture