Why simplices might be the most spread-out convex bodies

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Family 101 · Convex and metric geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Choose a point uniformly at random inside a convex body. Its covariance matrix records how far the point typically lies from the center in every direction. Some bodies spread their volume far from the center; others concentrate it more tightly. Which shape produces the greatest spread once changes of scale and linear coordinates have been factored out?

The relevant quantity is the isotropic constant. A long-standing sharp conjecture proposes that a simplex maximizes it in every dimension. A triangle is the two-dimensional simplex, a tetrahedron the three-dimensional one, and the general simplex is the convex hull of $n+1$ affinely independent points in $\mathbb R^n$.

Family 101 claims this exact inequality and its equality classification. Its route goes through a broader probabilistic statement: among log-concave distributions with fixed covariance, products of one-sided exponential distributions have the smallest differential entropy, after allowing invertible affine changes of coordinates.

The bridge between the two problems is particularly revealing. A convex body becomes the base of a cone in one higher dimension; an exponential density on that cone converts geometric spread into an entropy comparison.

2. Background and essential definitions

Measuring spread without choosing coordinates

Let $K\subset\mathbb R^n$ be compact and convex with nonempty interior, and let $|K|$ denote its volume. Its centroid and covariance matrix are

$$b_K=\frac1{|K|}\int_Kx\,dx, \qquad \Sigma_K=\frac1{|K|}\int_K(x-b_K)(x-b_K)^T\,dx.$$

Here a vector is a column, and the superscript $T$ means transpose. For any vector $v$, the number $v^T\Sigma_Kv$ is the variance of the coordinate $v^TX$ of a uniform random point $X$ in $K$. The determinant combines the variances along the covariance matrix’s orthogonal principal directions.

Define

$$L_K=\left(\frac{\det\Sigma_K}{|K|^2}\right)^{1/(2n)}.$$

An invertible linear transformation $A$ multiplies the covariance determinant by $(\det A)^2$ and the squared volume by the same factor. Translation changes neither. Thus $L_K$ depends on shape up to invertible affine transformation. All nondegenerate simplices in a fixed dimension have the same value.

In coordinates where $|K|=1$, the centroid is zero, and $\Sigma_K=L_K^2I$, the body is in isotropic position. The identity matrix $I$ indicates equal directional variances. This terminology is a normalization, not an assumption that the body has rotational symmetry.

Entropy and convexity of a density

A probability density $f$ on $\mathbb R^m$ is log-concave if it can be written $f=e^{-V}$ for a convex function $V$, allowing $V=+\infty$ outside its support. Uniform densities on convex bodies and one-sided exponential densities are examples. Convexity excludes separate peaks with a deep valley between them.

Its differential entropy is

$$h(f)=-\int_{\mathbb R^m}f(x)\log f(x)\,dx,$$

using natural logarithms and $0\log0=0$. Unlike discrete entropy, this quantity can be negative and changes with the units of measurement. The affine-invariant combination relevant here is

$$h(f)-\frac12\log\det\operatorname{Cov}(f).$$

Both terms change by $\log|\det A|$ under $x\mapsto Ax+b$, so their difference does not change. The densities in the claimed theorem are with respect to full-dimensional Lebesgue measure; their covariance matrices are positive definite.

3. Development history and previous partial results

Bourgain’s slicing problem asks whether $L_K$ has a universal upper bound independent of dimension. Its geometric formulation concerns large hyperplane sections of volume-one convex bodies. Controlling covariance and controlling sections turn out to be closely related.

Klartag obtained an $O(n^{1/4})$ upper bound. Eldan’s stochastic localization method later opened a route to subpolynomial and polylogarithmic bounds through work including Chen and Klartag–Lehec. The manuscript’s historical baseline includes Klartag and Lehec’s 2025 dimension-independent result using Guan’s covariance estimate.

Consequently, this family’s claimed advance must be distinguished from establishing that some universal constant exists. It proposes the exact best constant in each dimension and identifies every extremal body.

In dimension two, Campi, Colesanti, and Gronchi proved the simplex bound, while Saroglou established the uniqueness of the maximizing shape. In higher dimensions, Rademacher showed that a simplicial polytope maximizing the isotropic constant among all convex bodies must be a simplex. A simplicial polytope has simplex-shaped facets; the restriction did not cover arbitrary convex bodies or show that every maximizer has that structure.

The entropic side also has its own history. Bobkov and Madiman developed the relation between entropy bounds for log-concave laws and slicing. Fradelizi and Marín Sola formulated the sharp entropy inequality used here and proved its equivalence, across dimensions, with the simplex inequality. Melbourne, Nayar, and Roberto established the sharp one-dimensional entropy bound, with a one-sided exponential law as extremizer.

The new manuscript therefore builds on both geometric partial results and an established equivalence. Its claimed new work is the all-dimensional sharp entropy inequality, including the equality analysis needed to recover unique simplex extremizers.

4. What the new manuscripts claim

The exact simplex bound

The family has one manuscript. Its first theorem states that every convex body in every dimension $n\geq1$ satisfies

$$L_K\leq \frac{(n!)^{1/n}} {(n+1)^{(n+1)/(2n)}\sqrt{n+2}},$$

with equality if and only if $K$ is a simplex. Here $n!=1\cdot2\cdots n$. No symmetry, smooth boundary, or polytope assumption appears.

In dimension one, every convex body is an interval, hence a simplex, and the formula gives $1/\sqrt{12}$. In higher dimensions the classification says all affine shapes other than simplices give a strict inequality. The right side tends to $1/e$ as the dimension grows; the statement is much more precise than a dimension-independent upper estimate.

A sharp entropy inequality and all equality cases

The second theorem claims, for every log-concave probability density on $\mathbb R^m$,

$$h(f)\geq m+\frac12\log\det\operatorname{Cov}(f).$$

Equality holds precisely for distributions of

$$M(E_1,\ldots,E_m)^T+b,$$

where $M$ is invertible, $b$ is a vector, and the $E_i$ are independent variables with density $e^{-u}$ for $u\geq0$, and zero otherwise. Each has mean, variance, and entropy equal to one, so their product attains the bound.

With covariance $I$, the inequality reads $h(f)\geq m$. It is a lower bound restricted to log-concave densities. The familiar Gaussian maximum-entropy theorem is an upper bound at fixed covariance and has a different extremizer.

A consequence for volume products

Through an earlier implication of Klartag, the claimed sharp simplex bound also yields the nonsymmetric Mahler inequality

$$|K|\,|K^\circ|\geq\frac{(n+1)^{n+1}}{(n!)^2}$$

when the origin lies in the interior of $K$. The polar is $K^\circ=\{y:\langle x,y\rangle\leq1\text{ for all }x\in K\}$. This is a further route to a claim discussed in family 087. The consequence stated here is the inequality; the manuscript’s principal equality classifications concern entropy and isotropic constants.

5. Why the advance matters

Provisional importance: 5/5. The claim settles a sharp central problem in asymptotic convex geometry, identifies its extremizers, and establishes a corresponding distributional inequality with a broader domain than uniform measures on bodies.

An exact extremal theorem explains which geometry makes a general bound tight. Here the simplex would be the most demanding shape for isotropic spread, while independent one-sided exponentials would be the most entropy-efficient log-concave building blocks at fixed covariance.

The implication for Mahler volume also connects two major affine-invariant quantities: covariance divided by volume, and a body’s volume multiplied by its polar’s volume. The importance rating reflects that mathematical reach if the claims are correct, not an assessment of their verification status.

6. The main idea

First solve a distributional problem

The manuscript begins with a smooth density whose negative logarithm has Hessian bounded above and below by positive constants. It transports a standard Gaussian distribution to the target density by the gradient of a convex function. The derivative of this map is a positive definite symmetric matrix at each point.

The one-dimensional model is explicit. If $\Phi$ is the standard Gaussian distribution function, then

$$t(a)=-\log\Phi(-a)$$

takes a standard Gaussian variable to a mean-one exponential variable. Let $q=t'$. Rather than compare the transport derivative directly with a constant matrix, the manuscript reparametrizes it using the inverse of $q$, applied to its eigenvalues.

A carefully chosen affine normalization makes this new matrix field have Gaussian mean zero. This is a nonlinear condition and is not supplied merely by setting the target covariance equal to the identity.

The field is then separated into its part linear in the Gaussian coordinates and its higher-degree part. Gaussian energy inequalities have equality in the linear directions, so those directions require separate handling. Higher-degree terms carry an energy surplus that can control the entropy deficit.

A major technical issue is that matrix derivatives need not commute. The argument retains terms measuring differences between eigenvalues and uses them to absorb off-diagonal errors. Scalar estimates and this matrix energy accounting together are claimed to make the entropy deficit nonpositive.

Equality must survive approximation

The expected extremizers vanish outside a cone and have affine potentials on their support. They do not belong to the initial smooth, uniformly convex class. The manuscript therefore approximates general log-concave densities while preserving entropy and covariance.

A quantitative remainder forces the reparametrized derivative of a nearly extremizing transport toward a linear matrix field. Compatibility with being a derivative then forces the constant coefficient matrices to commute. Simultaneous diagonalization reduces the limiting transport to independent one-dimensional exponential transports. This is how the equality classification emerges, rather than being guessed from the sharp constant alone.

Put a convex body under an exponential roof

To return to geometry, translate $K$ so its centroid is zero and form the cone

$$C_K=\{(tx,t):x\in K,\ t\geq0\}\subset\mathbb R^{n+1}.$$

Its section at height $t$ is a copy of $K$ scaled by $t$. On this cone, use the density

$$f_K(y,t)=\frac{e^{-t}}{n!|K|}\,\mathbf1_{C_K}(y,t),$$

where the indicator is one on the cone and zero elsewhere. The normalization follows from $\int_0^\infty t^ne^{-t}dt=n!$.

The manuscript gives

$$\begin{aligned} h(f_K)&=n+1+\log(n!|K|),\\ \det\operatorname{Cov}(f_K)&=(n+1)^{n+1}(n+2)^n\det\Sigma_K. \end{aligned}$$

Substitution into the entropy inequality in dimension $n+1$ gives exactly the claimed simplex constant. If equality holds, the cone’s density must be an affine image of a product of exponentials. Its support is consequently a cone with $n+1$ independent generating rays. Intersecting those rays with the height-one plane makes $K$ a simplex.

7. Scope, limitations, and open questions

The entropy theorem concerns log-concave densities on full-dimensional Euclidean space. It does not give the same lower bound for arbitrary densities with a prescribed covariance, or for discrete distributions with Shannon entropy. Changing the logarithm convention would also change the numerical constants.

Smoothness and uniform curvature belong to an intermediate stage, not the final hypothesis. Uniform distributions on arbitrary convex bodies are included, and the claimed equality distributions have a boundary to their support.

The equality classification is exact. A separate geometric stability question asks how close a body must be to a simplex when its isotropic constant nearly attains the bound, with an explicit distance and dimension dependence. The manuscript’s analytic remainder and equality argument should not be read as already providing every such quantitative stability theorem.

The result also does not classify maximizers under extra symmetry constraints. A centrally symmetric body in dimension greater than one cannot itself be a simplex, so sharp bounds in restricted classes are distinct questions.

8. Sources and further reading

For the preceding dimension-independent result, see Klartag and Lehec, Affirmative resolution of Bourgain’s slicing problem using Guan’s bound, and Guan, A note on Bourgain’s slicing problem. Rademacher’s simplicial-polytope theorem records an earlier restricted extremal result.

For the entropic development, see Bobkov and Madiman, The entropy per coordinate of a random vector is highly constrained under convexity conditions; Fradelizi and Marín Sola, Entropy, slicing problem and functional Mahler’s conjecture, Indiana University Mathematics Journal 75 (2026), 825–843; and Melbourne, Nayar, and Roberto, Minimum entropy of a log-concave variable with fixed variance. Klartag’s Isotropic constants and Mahler volumes supplies the earlier implication for polar volume products.

Source manuscripts

• M1. OpenAI, A sharp entropy bound and the simplex inequality for isotropic constants (October 5, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.