Strong Hyperplane Conjecture for Isotropic Constants

Strong Hyperplane Conjecture for Isotropic Constants

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 101; 2026-10-05), the asserted result is that simplices uniquely maximize the isotropic constant, and log-concave densities satisfy the stated sharp entropy--covariance bound with affine exponential-product equality.

Mathematical statement of the claimed result

Manuscript 1.

For every $n\ge1$ and every full-dimensional convex body $K\subset\mathbb R^n$, use $V=|K|$, its centroid $b$, covariance $\Sigma_K=V^{-1}\int_K(x-b)(x-b)^{\mathsf T}\,dx$, and $L_K=(\det\Sigma_K/V^2)^{1/(2n)}$. The claimed sharp bound, with equality exactly for simplices, is \[ L_K\le\frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)}\sqrt{n+2}}. \] The associated entropy assertion concerns every log-concave probability density $f$ on $\mathbb R^m$, $m\ge1$: \[ h(f)=-\int f\log f\ge m+\tfrac12\log\det\operatorname{Cov}(f). \] Equality is claimed precisely for affine images $M(E_1,\ldots,E_m)^{\mathsf T}+b$, with $M$ invertible and independent standard exponential variables $E_i$; densities are identified almost everywhere.

Outline of the argument

For manuscript 1 (family 101; 2026-10-05), gaussian decomposition expands the entropy deficit into weighted chaos terms and commutator squares. Certified scalar inequalities control the expansion, and vanishing slack forces an exponential-product law; applying the bound to uniform bodies yields simplex rigidity.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 101; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, A sharp entropy bound and the simplex inequality for isotropic constants, 2026-10-05; citation key OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: intro:simplex, intro:simplex-bound.
• OpenAI catalogue entry.

Strong Hyperplane Conjecture for Isotropic Constants

Definitions and mathematical statement

For a convex body $K\subset{\mathbb{R}}^n$, let $V=|K|$, $b=V^{-1}\int_Kx{\,\mathrm{d}} x$, and \[ \Sigma_K=\frac1V\int_K(x-b)(x-b)^{\mathsf T}{\,\mathrm{d}} x, \qquad L_K=\left(\frac{\det\Sigma_K}{V^2}\right)^{1/(2n)}. \] The covariance is positive definite because $K$ has nonempty interior. This definition is invariant under invertible affine changes of coordinates. The sharp conjecture is \[ L_K\le L_{\Delta_n} =\frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)}\sqrt{n+2}}, \] where $\Delta_n$ is any nondegenerate simplex. It applies in each dimension $n\ge1$, without symmetry or smoothness assumptions. It seeks the simplex's precise dimension-dependent bound, not merely some universal upper bound or a uniqueness statement for equality.

Formulation and concept references: [S1].

Short English statement

In every dimension, does a simplex maximize the affine-invariant isotropic constant among all convex bodies?

Research attempt

Which hyperplane conjecture is being tested.

The statement fixes a dimension-dependent extremal value. It must not be confused with the existence of some dimension-free upper bound. The inspected revision [S1], dated 27 May 2025, explicitly distinguishes the latter result from the strong simplex conjecture. Its Theorem 1.2 concerns polytopal local maximizers with a simplicial vertex; it does not say every maximizer has such a vertex. We retain that distinction. The arguments below compute several model families, derive a first-variation obstruction, and explain a concrete failure of a proposed reduction by approximation.

Write $F(K)=L_K^{2n}=\det\Sigma_K/|K|^2$. If $T(x)=Ax+c$ with $A$ invertible, then \[ |T(K)|=|\det A||K|,\quad b_{T(K)}=Ab_K+c, \quad\Sigma_{T(K)}=A\Sigma_KA^{\mathsf T}. \] Consequently $F(T(K))=F(K)$. In particular, translating by the barycenter and applying $\Sigma_K^{-1/2}$ puts the body in the normalization $b=0$, $\Sigma=I$. In this normalization \[ L_K=|K|^{-1/n},\qquad \int_K|x|^2\,dx=n|K|. \tag{358.1} \] This is covariance-one normalization, rather than the alternative volume-one normalization sometimes also called isotropic position.

The simplex value, including its determinant.

Take the coordinate simplex $S=\{x_i\geq0:\sum_{i=1}^nx_i\leq1\}$. Repeated integration, or induction using the last coordinate, gives \[ \int_Sx_1^{a_1}\cdots x_n^{a_n}\,dx =\frac{\prod_{i=1}^na_i!}{(n+\sum_i a_i)!} \qquad(a_i\hbox{ nonnegative integers}). \tag{358.2} \] For the induction, integrate the first $n-1$ coordinates over a simplex of scale $1-x_n$ and use $\int_0^1t^a(1-t)^b\,dt=a!b!/(a+b+1)!$, which itself follows by repeated integration by parts. Thus $358.2$ does not assume a probability formula for simplex coordinates.

It follows that $|S|=1/n!$, $b_i=1/(n+1)$ and \[ \mathbb E X_i^2=\frac2{(n+1)(n+2)},\qquad \mathbb E X_iX_j=\frac1{(n+1)(n+2)}\ (i\ne j). \] Subtracting the means gives \[ \Sigma_S=\frac{(n+1)I-J}{(n+1)^2(n+2)}, \qquad \det\Sigma_S=\frac1{(n+1)^{n+1}(n+2)^n}, \tag{358.3} \] where $J$ is the all-ones matrix. To check the determinant, the numerator has eigenvalue one in the all-ones direction and eigenvalue $n+1$ on its orthogonal complement. Combining $358.3$ with the volume recovers exactly the printed simplex constant. All nondegenerate simplexes have the same value by affine invariance.

For $n=1$ every convex body with interior is an interval. Its length $\ell$ and variance $\ell^2/12$ give $L=1/\sqrt{12}$, so the selected conjecture is proved completely in dimension one, including all translations and lengths.

Product bodies and two comparison families.

For full-dimensional $K\subset\mathbb R^a$ and $H\subset\mathbb R^b$, the uniform random point of $K\times H$ has independent components. Its covariance is block diagonal and its volume is the product. Therefore \[ L_{K\times H}^{a+b}=L_K^aL_H^b. \tag{358.4} \] In particular every parallelepiped has $L=1/\sqrt{12}$. Product structure averages logarithms of isotropic constants; it cannot produce a value larger than both factors. But this identity does not reduce arbitrary bodies to products: most bodies have no such affine decomposition.

For the Euclidean unit ball $B_2^n$, write its volume as $\kappa_n$. Rotational invariance gives a scalar covariance matrix, and radial integration gives $\mathbb E|X|^2=n/(n+2)$. Hence \[ L_{B_2^n}=\frac1{\sqrt{n+2}\,\kappa_n^{1/n}}. \tag{358.5} \] For the cross-polytope $B_1^n=\{\sum_i|x_i|\leq1\}$, the $2^n$ orthants are reflected coordinate simplexes. Reflection makes all means and mixed second moments vanish, while $358.2$ gives $\mathbb E X_i^2=2/((n+1)(n+2))$ and $|B_1^n|=2^n/n!$. Thus \[ L_{B_1^n}=\frac{(n!)^{1/n}}{\sqrt{2(n+1)(n+2)}}. \tag{358.6} \] The ratio to the simplex value is \[ \frac{L_{B_1^n}}{L_S} =\frac{(n+1)^{1/(2n)}}{\sqrt2}\leq1, \tag{358.7} \] because $n+1\leq2^n$ for $n\geq1$, by elementary induction. This proves the desired comparison for every affine cross-polytope. It is a full family calculation, not a numerical comparison in low dimensions. In dimension two the cross-polytope is a parallelogram, consistent with $358.4$.

An extremal statement in the opposite direction.

There is a direct proof that ellipsoids minimize this functional. It is worth spelling out because a moment-minimization argument can otherwise be accidentally used with the wrong sign.

Normalize an arbitrary $K$ as in $358.1$ and set $V=|K|$. Let $D$ be the centered ball of volume $V$, with radius $r=(V/\kappa_n)^{1/n}$. Since $|K\setminus D|=|D\setminus K|$, \[ \int_K|x|^2-\int_D|x|^2 =\int_{K\setminus D}(|x|^2-r^2) +\int_{D\setminus K}(r^2-|x|^2)\geq0. \] Using $\int_K|x|^2=nV$ and $\int_D|x|^2=nVr^2/(n+2)$ gives $r^2\leq n+2$. Therefore \[ V\leq\kappa_n(n+2)^{n/2},\qquad L_K\geq\frac1{\sqrt{n+2}\kappa_n^{1/n}}. \tag{358.8} \] Equality forces $K=D$ up to measure zero, and convex bodies with interior agreeing almost everywhere agree as closed sets. Undoing the affine normalization proves the ellipsoid equality statement. The inequality points downward for $L_K$. Rearrangement supplies the minimum, whereas the question asks for the maximum.

First variation under a genuine smooth deformation.

Suppose temporarily that $K$ has a smooth boundary with strictly positive principal curvatures. Normalize $b=0$ and $\Sigma=I$. Let $K_t$ be a smooth normal deformation with scalar normal velocity $h$ on the boundary. Denote volume, first moment and second moment by $V(t)$, $M(t)=\int_{K_t}x\,dx$, $Q(t)=\int_{K_t}xx^{\mathsf T}\,dx$. Differentiating integrals over a moving boundary gives \[ V'=\int_{\partial K}h\,dS,\quad M'=\int_{\partial K}xh\,dS,\quad Q'=\int_{\partial K}xx^{\mathsf T}h\,dS. \] These formulas follow by integrating the thickness $th+o(t)$ of the boundary layer in local normal coordinates. Since $M(0)=0$, the derivative of $bb^{\mathsf T}$ is zero at $t=0$. As $Q(0)=VI$, \[ \Sigma'=V^{-1}Q'-(V'/V)I. \] Jacobi's formula for the determinant now gives \[ \left.\frac{d}{dt}\right|_{0}\log F(K_t) =\frac1V\int_{\partial K}(|x|^2-n-2)h(x)\,dS(x). \tag{358.9} \] The centering correction has been included rather than silently held fixed; it drops out only because its first variation vanishes at zero mean. The coefficient $n+2$ comes from both covariance normalization and the squared volume in $F$.

Every smooth $h$ is available in both signs for sufficiently small $|t|$ under the positive-curvature assumption. One way to see this is to perturb the support function on the sphere: its curvature matrix is positive definite, and positivity survives a sufficiently small $C^2$ perturbation. The Gauss map identifies the support-function velocity with the desired normal velocity.

If such a body were a local maximizer, $358.9$ would vanish for every $h$, forcing $|x|^2=n+2$ on its boundary. Since a centered convex body has zero in its interior, every ray meets its boundary once; the constant radius then forces it to be the ball of radius $\sqrt{n+2}$. For $n\geq2$ that ball is not a local maximum: $358.8$ is strict for every nearby nonellipsoid. Such nearby strictly convex bodies exist, for example through a small smooth support perturbation outside the finite-dimensional family of ellipsoids. We have therefore proved: \[ \text{A smooth positively curved body in dimension }n\geq2 \text{ cannot locally maximize }L_K. \tag{358.10} \] This does not exclude the conjectured simplex; its boundary is outside the regularity and two-sided deformation assumptions of the argument.

Why this variation does not classify nonsmooth maximizers.

At a flat facet, an arbitrary normal displacement need not preserve convexity for both signs. At an edge or a vertex the admissible variations are more constrained still. Applying $358.9$ as though all boundary velocities were freely available would incorrectly rule out polytopes, including the simplex. One must characterize a cone of admissible variations and derive one-sided inequalities on that cone. The source's polytopal theorems use additional geometric structure; their hypotheses cannot be deleted by the smooth calculation above.

An approximation argument does not repair this. If convex bodies $K_j$ converge to a full-dimensional $K$ in Hausdorff distance, their volumes and first two moments converge. Indeed they lie in a common bounded set and their indicators converge almost everywhere off $\partial K$, which has Lebesgue measure zero; dominated convergence then applies to the bounded coordinate monomials. Thus $L_{K_j}\to L_K$. This allows upper bounds valid for every approximating polytope to pass to the limit. A theorem only about polytopes that locally maximize $L$ is different: an arbitrary polytope in an approximating sequence need not be a local maximizer. Smoothing has the same problem in reverse, as $358.10$ illustrates.

Likewise, triangulating a polytope does not express its covariance as a volume-weighted average of the simplex covariance determinants. For a disjoint union with component masses $w_i$, means $b_i$ and covariances $\Sigma_i$, direct expansion gives \[ \Sigma=\sum_iw_i\Sigma_i+ \sum_iw_i(b_i-b)(b_i-b)^{\mathsf T}. \tag{358.11} \] The positive between-component term is essential. The determinant is nonlinear, and omitting this term would already give a wrong answer for two adjacent intervals. Thus known simplex moments do not prove the upper bound for a triangulated body.

Diagnostics and outcome.

The accompanying exact checks evaluate the simplex moment matrix and determinant, the cross-polytope ratio, and the product identity in low dimensions. They test normalization and algebra only; no finite list of shapes certifies a supremum over all convex bodies. The actual proved results are the simplex value, the complete one-dimensional case, the affine cross-polytope comparison, the ellipsoid minimum, and the smooth positive-curvature obstruction.

The first unresolved step toward the selected sharp upper bound is controlling possible nonsmooth maximizers without adding a simplicial vertex, product decomposition, or unrestricted deformation assumption. The universal slicing bound does not supply this sharp comparison. The conjecture remains unresolved by this attempt.

Sources

• S1 Isotropic constants and regular polytopes; arXiv2407.01353v2,27May2025. https://arxiv.org/pdf/2407.01353v2. Formulation source.