Projection-body extrema: ellipsoid minima and product counterexamples

Source and verification

This article explains family 088 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Projection bodies and affine invariance

For a convex body $K\subset {\mathbb{R}}^n$, the support function is $h_K(u)={\sup }_{x\in K}⟨x,u⟩$. Its projection body $\Pi K$ is the origin-symmetric convex body characterized, for unit vectors $u$, by

$$h_{\Pi K}(u)=|{\mathrm{proj}}_{u^{\perp }}K|.$$

The right-hand side is an $(n-1)$-dimensional volume. Extend it homogeneously to other vectors. The functional in this family is

$$R_n(K)=\frac{|\Pi K|}{|K{|}^{n-1}}.$$

An invertible linear map satisfies $\Pi (TK)=|\det T|T^{-T}\Pi K$. Taking determinants shows that both numerator and denominator acquire the factor $|\det T{|}^{n-1}$. Thus $R_n$ is affine invariant; translations also have no effect.

The two manuscripts investigate opposite ends of this functional. One claims the sharp lower bound and ellipsoid equality cases. The other gives a product construction exceeding the simplex value. (OpenAI, 2026) (OpenAI, 2026a)

🏷️ The proposed minimum

Write ${\kappa }_n=|B_2^n|$. The shadow of the unit ball has volume ${\kappa }_{n-1}$ in every direction, so

$$\begin{array}{c}\Pi B_2^n={\kappa }_{n-1}{B}_2^n,\\ R_n(B_2^n)={\kappa }_{n-1}^n{\kappa }_n^{2-n}.\end{array}$$

Statement in the minimization manuscript

For every convex body $K\subset {\mathbb{R}}^n$ with $n\ge 4$,

$$R_n(K)\ge {\kappa }_{n-1}^n{\kappa }_n^{2-n},$$

with equality exactly for ellipsoids. Symmetry and smoothness of $K$ are not assumed. (OpenAI, 2026b)

This is the unpolarized projection-body minimization problem. The classical inequality involving the polar of $\Pi K$ is a different functional. The manuscript credits a separate external result for dimension three; its stated new dimensional range begins at four.

🏷️ First variations and the global argument

Minkowski addition is defined by $K+tM=\{x+ty:x\in K,y\in M\}$. The first derivative of volume under this operation is expressed by the surface-area measure:

$${\frac{d}{dt}|K+tM||}_{t=0}={\int }_{{\mathbb{S}}^{n-1}}{h}_{M}d{S}_K.$$

This representation converts a variation of a body into a pairing between a support function and a measure. Projection bodies introduce the kernel $|⟨x,y⟩|$, so the manuscript’s central comparison becomes a sharp bilinear inequality between normalized first-variation measures. (OpenAI, 2026b)

The normalization is selected so that each measure dominates the appropriate volume functional on support functions. A harmonic decomposition then separates changes corresponding to affine coordinates from higher-order shape changes. Degrees at least four carry a strict deficit. Degree two has to be treated through affine normalization; it cannot simply be discarded as a zero mode.

Local information near the ball would not establish a global minimum. The source supplements the harmonic estimate with a variational construction on matrix parameters. Its argument must remain continuous as matrices approach singularity and must rule out a minimizing configuration on that boundary. An inward-pointing field and a fixed-point argument supply this global step. Finally, equality in the first-variation comparisons forces the original body to be an ellipsoid.

This is the part of the paper that a full verification would need to inspect beyond the sharp scalar constant: the passage from an admissible measure to an actual body, the singular parameter boundary, and the recovery of equality all carry geometric content.

🏷️ Products and the failure of simplex maximality

Let $A\subset {\mathbb{R}}^r$ and $B\subset {\mathbb{R}}^s$. The product identity used in the second manuscript is

$$R_{r+s}(A\times B)=R_r(A)R_s(B).$$

For polytopes, the facets of a product come in two kinds, inherited from $A$ and from $B$. Their surface-area contributions give

$$\Pi (A\times B)=(|B|\Pi A)\times (|A|\Pi B).$$

Taking volume yields the multiplicativity after division by $(|A||B|{)}^{r+s-1}$. Approximation gives the identity for general convex bodies.

If $T_d$ is a $d$-simplex, its affine-invariant value is

$$c_d=R_d(T_d)=\frac{(d+1)d^d}{d!}.$$

The product $T_{10}\times T_{10}$ therefore gives the explicit comparison

$$\begin{array}{rl}\frac{R_{20}(T_{10}\times T_{10})}{R_{20}(T_{20})}& =\frac{121\left(\genfrac{}{}{0ex}{}{20}{10}\right)}{212^{20}}\\ & =\frac{22355476}{22020096}>1.\end{array}$$

The inequality is exact integer arithmetic. It disproves simplex maximality for this functional in dimension twenty, provided the displayed geometric identities. The source also uses repeated products for an exponentially growing excess in sufficiently large dimensions. (OpenAI, 2026a)

The manuscript credits earlier counterexamples by Feng beginning in dimension nine. Its contribution is an explicit product mechanism and amplification, rather than the first negative answer to simplex maximality. The maximum-side counterexample has no conflict with the proposed ellipsoid minimum: the two optimize in opposite directions.

🏷️ Key steps and proof outlines

First variations replace shapes by normalized measures

A perturbation of a support function changes volume through its surface-area measure. The projection-body support function itself is a cosine transform of that measure: it integrates $|⟨u,v⟩|$ against surface normals, with the conventional factor from the projection formula. Applying the volume variation again therefore produces a bilinear expression involving two measures and the absolute inner-product kernel.

The source normalizes the measures so their pairings with support functions dominate a specified volume functional. This is a geometric admissibility condition, stronger than merely having the correct total mass. Its sharp bilinear theorem can then be applied to the measures generated by the body and its projection body. The normalization constants are chosen so the ball realizes equality. See “Gauges and projection measures” and “The bilinear inequality and equality cases.” (OpenAI, 2026b)

Spherical harmonics distinguish affine motion from shape deficit

The kernel $|⟨u,v⟩|$ is rotation invariant, so its integral operator acts by scalar multipliers on each spherical-harmonic degree. Odd components are invisible to this even kernel. Degree zero controls the mean, and degree two corresponds to infinitesimal changes of ellipsoidal shape induced by linear coordinates. Higher even degrees represent deviations that affine normalization cannot remove.

The source proves a strict sign or spectral estimate on these higher degrees, then combines it with a convexity inequality for norms. The proof strategy is to decompose the normalized data, identify the exact low-degree terms, and show the remaining terms carry a positive deficit. A spectral estimate near a ball would yield only local information. The manuscript uses it inside a variational representation that applies to general admissible data.

The degree-two contribution is particularly important: it cannot simply be set to zero by declaring an arbitrary body isotropic. The affine-position construction has to select a matrix that enforces the normalization required by the bilinear argument. The norm inequality and matrix selection are what connect the harmonic calculation back to the original body.

Boundary control makes the affine normalization global

The parameter space of normalizing matrices has a boundary consisting of singular matrices. A minimizing sequence might approach that boundary and flatten some directions. The source extends the relevant quantities continuously to an appropriate compactified parameter set and proves an inward-pointing property of a variational field. A fixed-point argument then gives an interior normalization.

This step performs two jobs. It establishes existence of the needed position without assuming a priori bounds on eccentricity, and it rules out a degenerate configuration that could evade the strict harmonic estimate. Equality is subsequently traced through both the variational representation and the bilinear comparison. The strict higher-degree deficit leaves only ellipsoidal data, and the first-variation equalities identify the original body as an ellipsoid.

The maximum-side construction uses facets instead of variations

For a polytope $P$, its projection body is a zonotope: a Minkowski sum of symmetric segments determined by facet normals weighted by facet areas. A facet of $A\times B$ is either $F_A\times B$ or $A\times F_B$. Its area is respectively $|F_A||B|$ or $|A||F_B|$, and its normal lies entirely in the corresponding coordinate factor. Thus the zonotope splits into complementary factors,

$$\Pi (A\times B)=(|B|\Pi A)\times (|A|\Pi B).$$

Taking volumes gives factors $|B{|}^r$ and $|A{|}^s$. Dividing by $(|A||B|{)}^{r+s-1}$ leaves exactly $R_r(A)R_s(B)$. This checks every exponent in multiplicativity. Hausdorff approximation by polytopes and continuity of projection bodies extend the identity to arbitrary convex bodies. (OpenAI, 2026a)

For a simplex, the determinant formula for zonotope volume reduces the calculation to its facet vectors. Their unique linear relation and equal determinant magnitudes give $c_d=(d+1)d^d/d!$. Substituting two ten-dimensional simplices yields the exact positive integer difference in the displayed dimension-twenty comparison. Repeating products amplifies the advantage because product values multiply, while the simplex benchmark has different asymptotics. This is a constructive global obstruction to simplex maximality, logically independent of the harmonic and variational minimum argument.

See Also

Mahler volume products introduces polarity and contrasts symmetric and nonsymmetric affine extrema. Covering densities also separates a universal bound from examples witnessing the worst-case order.

🏷️ Manuscripts and verification scope

• Petty’s projection-volume conjecture in dimensions at least four — pinned manuscript and companion source.
• A product counterexample to the simplex maximum for projection-body volume — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope covers the minimum in dimensions at least four and its ellipsoid equality statement, and the explicit dimension-twenty product witness. It does not include the product manuscript’s full exponential amplification. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A product counterexample to the simplex maximum for projection-body volume.

🐻  OpenAI 2026b. Petty’s projection-volume conjecture in dimensions at least four.