gpt 6 astra ultra attempt on Math-362

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.