The sharp simplex conjecture for isotropic constants

In every dimension, simplices alone have the largest isotropic constant among convex bodies, the paper claims.

1 paper October 5, 2026 Explained by AI

The isotropic constant compares how spread out a random point of a solid shape is with the shape's volume. The strong isotropic constant conjecture says the simplex, such as a triangle or tetrahedron, gives the largest value. This paper claims to prove it in every dimension, with simplices as the only maximizers. It also claims a sharp lower bound on the entropy of log-concave distributions. No Lean formalization is listed.

First step

1986

Jean Bourgain, the slicing problem

Significance

Major

If the proof holds, it finds the exact extreme shape for a central quantity in convex geometry, in every dimension.

Trust

Claimed, not checked

Field

Convex and metric geometry

1 paper

The explanation

Take three flat shapes: a square, a disk and a triangle. Drop a random point uniformly inside each. Measure how spread out that point is, using its covariance [the table of variances in each direction]. Then compare the spread with the area. The isotropic constant L packages this comparison into one numberSource 1.

Here are the values, computed exactly. The square gets the square root of 1/12, about 0.2887. The disk gets 1/(2√π), about 0.2821. The triangle gets about 0.3102. The triangle wins. Stretching, turning or sliding a shape does not change its LSource 2. So every triangle gets the same 0.3102.

The paper claims the triangle pattern holds in every dimensionSource 3. Among all convex bodies [solid shapes with no dents], simplices have the largest isotropic constant. A simplex is the higher-dimensional triangle, such as a tetrahedron in three dimensions. The paper also claims simplices are the only maximizersSource 4. The simplex value is about 0.3102 in two dimensions and about 0.3225 in three.

The question goes back to the slicing problem, from Bourgain's 1986 workSource 5. It asked whether L stays below one fixed number in every dimension. Klartag proved a bound growing like the fourth root of the dimension, in 2006Source 6. Later work by many authors brought this down step by step. Klartag and Lehec then proved a bound independent of dimension, using Guan's estimateSource 7. But a universal bound does not name the largest value or the shape that reaches itSource 8.

The exact answer was known only in the plane. There Campi, Colesanti and Gronchi proved the triangle bound, and Saroglou proved the triangle is the only maximizerSource 9. In higher dimensions, Rademacher showed in 2016 that a maximizer with triangular faces must be a simplexSource 10. That ruled out some shapes but not all.

The paper's route goes through entropy. The differential entropy of a distribution measures how spread out and unpredictable it is. A log-concave density is one whose logarithm is concave, so it has one hump. The paper claims every such density in m dimensions has entropy at least m plus half the log of its covariance determinantSource 11. Equality holds only for stretched and shifted products of one-sided exponential lawsSource 12.

Check it in one dimension. The exponential law with mean 1 has entropy 1 and variance 1. The bound is 1 + ½ log 1 = 1, so it is met exactly. The uniform law on [0, 1] has entropy 0 and variance 1/12. Its bound is 1 + ½ log(1/12), about −0.2425, which 0 clears. The bell curve has entropy about 1.419, well above 1.

Melbourne, Nayar and Roberto had proved this one-dimensional caseSource 13. Fradelizi and Marín Sola stated the general bound and showed it matches the simplex bound across dimensionsSource 14. So proving the entropy bound in every dimension gives the simplex result. The hard part is equality. The expected extreme cases have cut-off supports, so smooth approximations alone cannot capture themSource 15. Part of the proof rests on explicit one-variable inequalities whose constants are exact fractionsSource 16.

There is a further consequence. Klartag showed in 2018 that the sharp simplex bound implies a nonsymmetric Mahler inequality, on a shape's volume times its polar's volumeSource 17. In two dimensions that floor is 27/4 = 6.75. A square from −1 to 1 has area 4, and its polar diamond has area 2. Their product, 8, sits above the floor.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not add anything new to the dimension-free slicing bound, which Klartag and Lehec had already proved.

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 only if the proof holds. It claims the exact extreme shape in every dimension, building on the already-proved dimension-free bound. No Lean formalization is listed and no specialist review is published.

Words used

Convex body: A bounded solid shape with no dents and a nonempty inside.
Isotropic constant: A number comparing the spread (covariance) of a uniform random point in a shape with the shape's volume.
Simplex: The higher-dimensional version of a triangle: a segment, triangle, tetrahedron, and so on.
Covariance: The table of variances and joint variations of a random point's coordinates.
Differential entropy: A number measuring how spread out and unpredictable a continuous distribution is.
Log-concave density: A probability density whose logarithm is concave, giving a single hump.
Polar body: The set of directions y with x·y at most 1 for every point x of the shape.

The original paper

A sharp entropy bound and the simplex inequality for isotropic constants (PDF, opens in a new tab)
October 5, 2026

All quoted lines (17)

The isotropic constant compares the covariance of a uniform distribution with the volume of its support.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
This quantity is unchanged by invertible affine maps.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
We prove the sharp inequality for arbitrary convex bodies and classify all equality cases.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
The slicing problem, arising in Bourgain's work on convex bodies, asks whether it admits an upper bound independent of dimension.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
Klartag proved an $O(n^{1/4})$ bound.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
Subsequent progress used stochastic localization, introduced by Eldan, leading through subpolynomial and polylogarithmic estimates to Klartag and Lehec's dimension-independent bound using Guan's covariance estimate.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
The distinction is substantial: a universal bound does not identify the extremal value or an extremizing body.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
In the plane, Campi, Colesanti, and Gronchi proved the simplex bound, and Saroglou established uniqueness of the maximizing body.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
Rademacher proved that a simplicial polytope maximizing the isotropic constant among all convex bodies must be a simplex.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
We also prove the sharp entropy bound $h(f)\ge m+\tfrac12\log\det\Cov(f)$ for every log-concave probability density on $\R^m$ , with equality precisely for invertible affine images of products of one-sided exponential laws.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
We also prove the sharp entropy bound $h(f)\ge m+\tfrac12\log\det\Cov(f)$ for every log-concave probability density on $\R^m$ , with equality precisely for invertible affine images of products of one-sided exponential laws.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
In dimension one, Melbourne, Nayar, and Roberto proved the sharp bound, attained by a one-sided exponential distribution.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
Fradelizi and Mar\'in Sola formulated the sharp entropy statement (…) and proved its equivalence, across dimensions, with the simplex bound.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
The expected extremizers have restricted support and affine potentials, so equality cannot be studied solely within the smooth, uniformly convex class.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
All numerical constants in the scalar inequalities are exact rationals.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text
Using exponential measures on convex cones, Klartag proved that the sharp simplex bound implies $|K|\,|K^\circ|\ge\frac{(n+1)^{n+1}}{(n!)^2},$ the nonsymmetric Mahler inequality.
A sharp entropy bound and the simplex inequality for isotropic constants Back to text

Other sources

096The Gaussian propeller conjecture in every dimensionConvex and metric geometry
091Logarithmic and L p Brunn–Minkowski inequalities and the B-conjectureConvex and metric geometry
087The Mahler conjectures, functional inequalities and polar-product symplectic widthConvex and metric geometry
073The Falconer distance conjectureReal and complex analysis
063The generalized Mukai conjectureAlgebraic and complex geometry

Browse Convex and metric geometry Named conjectures Open since the 1980s Claimed, not yet checked