# Sharp projection-body inequalities and a counterexample to simplex maximization

> In four or more dimensions, ellipsoids have the smallest shadow body for their volume. Plain-English explainer, Lean status and sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-088/
Last updated: 2026-10-11

**In four or more dimensions, ellipsoids have the smallest shadow body for their volume.**

**Trust:** Formal proof listed · **Papers:** 2 · **First paper:** September 24, 2026 · **Field:** Convex and metric geometry · **Entry:** 088

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Every solid casts shadows, and the projection body collects all their areas into one new shape. In 1971 Petty conjectured that, for a fixed volume, ellipsoids make this shape as small as possible. This paper proves that in every dimension from four up (Lean proof listed). A companion paper proves that a product of two ten-dimensional simplices beats the twenty-dimensional simplex. That rules out Brannen's guess that simplices give the largest value.

## At a glance

- **First step:** 1971 (C. M. Petty, the projection-volume conjecture)
- **Significance:** Major. If the proof holds, it settles a 1971 conjecture in every dimension from four up.
- **Trust:** Formal proof listed
- **Field:** Convex and metric geometry (2 papers)

## The explanation

Hold a cube with sides of length 1 up to the sun. Face-on, its shadow is a square of area 1. Turn it so the sun shines along a long diagonal. The shadow grows into a hexagon of area about 1.73. Every direction gives a shadow, and every shadow has an area.

Now build one new shape from all those numbers. In each direction, let it reach out exactly as far as that shadow is large. Geometers call this the projection body (the shape that records every shadow area). For the unit cube it is a cube of side 2, with volume 8. The paper studies a score: the volume of the projection body, divided by a power of the original volume. That division makes the score ignore size. It also ignores any stretching or shearing, so a ball and every ellipsoid (a stretched ball) score the same.

In three dimensions the ball scores about 7.40 and the cube scores 8. A tetrahedron, the simplest pointed solid, scores 18. In four dimensions the ball scores 1024/81, about 12.64. The four-dimensional cube scores 16. The simplest four-dimensional simplex scores 160/3, about 53.3. A simplex is the analogue of a triangle or tetrahedron.

In 1971 Clinton Petty conjectured that ellipsoids have the lowest score, in every dimension from three up. Nobody could rule out some odd shape scoring lower. Earlier results only compared the ball with shapes already close to it, after stretching. In 2026 Mielke-Sulz handled shapes with an axis of rotational symmetry. Also in 2026, Chen, Feng, Li, Xi and Xu proved the full three-dimensional case. Dimensions four and higher stayed open.

This paper proves that case: in every dimension from four up, ellipsoids have the lowest score[1]. They are the only shapes that reach it[2]. The shape need not be symmetric or smooth[3]. The paper says that, together with the separate three-dimensional proof, this settles Petty's conjecture in every dimension from three up. The paper itself does not use or check that three-dimensional proof[4].

The main obstacle was that older methods needed a shape near a ball. Near a ball, small errors can be controlled. A general shape gives no such foothold. The paper's answer is to choose, for any shape, a stretch that puts it in a balanced position[5,6]. Finding such a stretch takes a careful existence argument. In that position, a sharp estimate shows the ball cannot be beaten.

A second paper looks at the other end. In 1996 Brannen proposed that simplices have the highest score. Feng, Hu, Liu and Xu had already found counterexamples in every dimension from nine up, using perturbed cubes. This paper gives a simpler one. Take two ten-dimensional simplices and form their product, a twenty-dimensional shape. A product's score is the product of its two factors' scores[7]. Each ten-dimensional simplex scores about 30,313[8]. Squared, that is about 918.9 million. The twenty-dimensional simplex scores only about 905.1 million. The exact ratio is 22,355,476/22,020,096, about 1.015[9].

So the simplex loses by about 1.5 percent in dimension twenty. The paper also argues that the gap grows. In every large enough dimension, some product of simplices beats the simplex by a factor that grows exponentially with dimension[10]. Neither paper finds the true highest score or the shapes that reach it[11]. The first paper also derives further sharp inequalities, including the Lutwak–Petty family, but those lean on the separate three-dimensional proof[12].

How sure is this? A computer proof checker called Lean has checked a formal version of the lowest-score theorem. It covers every convex shape in dimension four and up. Its scope note states that ellipsoids are exactly the minimizers. Lean has also checked the twenty-dimensional counterexample. The exponential gap and the other inequalities are outside that check. No specialist review has been published.

**What it doesn't show.** This does not find the largest possible score or the shapes that reach it.

## 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. Lean proofs are listed for the four-plus-dimensional minimum and the dimension-twenty counterexample. The exponential gap and the derived inequalities are not machine-checked. The three-dimensional case rests on another group's preprint. No specialist review is published.

## Words used

- **Convex body:** A solid, bounded shape with no dents or holes, such as a ball, cube or pyramid.
- **Projection body:** A shape that reaches out, in each direction, exactly as far as the shadow cast in that direction is large.
- **Ellipsoid:** A ball that has been stretched or squeezed along some directions.
- **Simplex:** The simplest pointed shape in a given dimension: a triangle in two dimensions, a tetrahedron in three.
- **Normalized projection volume:** The projection body's volume divided by a power of the original volume, so size and stretching do not change it.
- **Counterexample:** A single case that shows a proposed general statement is false.

## The original papers (2)

- [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf) - September 24, 2026 · Formal proof listed
- [A product counterexample to the simplex maximum for projection-body volume](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf) - September 24, 2026 · Formal proof listed

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/088.md)

## Quoted lines from the paper (12)

[1] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> We prove the unpolarized conjecture in every dimension at least four.

[2] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> Equality holds if and only if $K=a+TB_2^n$ for some $a\in\R^n$ and some invertible linear map $T$.

[3] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> The theorem includes bodies without symmetry or boundary regularity.

[4] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> Combining it with the separately proved three-dimensional theorem of Chen et al. establishes the conjecture for every $n\ge3$; that external theorem is not used in our proof for $n\ge4$.

[5] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> The latter permits a global choice of affine coordinates, where proximity to the ball is unavailable.

[6] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> An explicit calculation in their missing directions gives an inward-pointing matrix field, and a fixed-point argument yields an invertible matrix at which the degree-two component vanishes.

[7] From [A product counterexample to the simplex maximum for projection-body volume](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf):

> The useful mechanism is a product test: for full-dimensional polytopes \(A\subset\R^r\) and \(B\subset\R^s\), with \(r,s\ge2\), \[ R_{r+s}(A\times B)=R_r(A)R_s(B). \]

[8] From [A product counterexample to the simplex maximum for projection-body volume](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf):

> For the fixed ten-dimensional block, exact arithmetic yields \[ c_{10}=\frac{17\,187\,500}{567},\qquad A:=\frac{c_{10}}{\beta^{10}} =\frac{1\,638\,400\,000\,000}{1\,336\,956\,340\,797}>1. \]

[9] From [A product counterexample to the simplex maximum for projection-body volume](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf):

> Then \[ \frac{R_{20}(K)}{c_{20}} =\frac{121\binom{20}{10}}{21\cdot2^{20}} =\frac{22\,355\,476}{22\,020\,096}>1. \]

[10] From [A product counterexample to the simplex maximum for projection-body volume](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf):

> Iterating our product identity gives an exponential excess over the simplex benchmark: for some fixed \(\lambda>1\) and every sufficiently large \(n\), a product of simplices \(K_n\subset\R^n\) satisfies \(R_n(K_n)\ge\lambda^n c_n\).

[11] From [A product counterexample to the simplex maximum for projection-body volume](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf):

> The optimal upper constant and its maximizing bodies remain separate questions.

[12] From [Petty’s projection-volume conjecture in dimensions at least four](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf):

> Theorem … and the separately proved three-dimensional result establish Petty's conjecture for every $n\ge3$.

## Other sources

- [OpenAI, “Petty’s projection-volume conjecture in dimensions at least four”, openai/math preprint (24 September 2026)](https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf): “We prove the unpolarized conjecture in every dimension at least four.”
- [OpenAI, “A product counterexample to the simplex maximum for projection-body volume”, openai/math preprint (24 September 2026)](https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf): “Thus \(K\) violates (…) in dimension twenty.”
- [S. Chen, Y. Feng, Y. Li, D. Xi and L. Xu, “Petty's conjectured projection inequality in dimension three”, preprint, MathSciDoc:2608.23001 (2026)](https://archive.ymsc.tsinghua.edu.cn/pacm_download/741/12780-pettyinequality.pdf)
- [Y. Feng, S. Hu, W. Liu and L. Xu, “On the reverse projection inequality”, preprint, MathSciDoc:2608.23002 (2026)](https://archive.ymsc.tsinghua.edu.cn/pacm_download/743/12781--2026.8.26.pdf)
- [N. S. Brannen, “Volumes of projection bodies”, Mathematika 43 (1996), 255–264](https://doi.org/10.1112/S002557930001175X)
- [F. Mielke-Sulz, “The Petty conjecture for convex bodies of revolution”, arXiv:2609.13517 (2026)](https://arxiv.org/abs/2609.13517v1)

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