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Math-2633 · mathematics · Geometry

Simplex maximum for projection-body volume

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $n\geq 3$ and convex body $K\subset R^{n}$, must $|\Pi K|/|K|^{n-1}$ be at most the corresponding value for an n-simplex? The projection body $\Pi K$ has support function $h_{\Pi K}(u)=vol_{n-1}(proj_{u\perp }K)$ for unit u. No symmetry or regularity is assumed.

Short English statement

For every $n\geq 3$ and convex body $K\subset R^{n}$, must $|\Pi K|/|K|^{n-1}$ be at most the corresponding value for an n-simplex? The projection body $\Pi K$ has support function $h_{\Pi K}(u)=vol_{n-1}(proj_{u\perp }K)$ for unit u. No symmetry or regularity is assumed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2633-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2633

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 088; 2026-09-24).

Simplex maximum for projection-body volume

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 088; 2026-09-24), the asserted result is that taking the Cartesian product of two dimension-ten simplices yields a body whose normalized projection volume exceeds the simplex value in dimension twenty, refuting the proposed maximum.

Mathematical statement of the claimed result

Manuscript 1.
For a $d$-dimensional convex body, define its projection body by $h_{\Pi K}(u)=\operatorname{vol}_{d-1}(\operatorname{proj}_{u^{\perp}}K)$ on unit vectors, and set $R_d(K)=|\Pi K|/|K|^{d-1}$ and $c_d=(d+1)d^d/d!$. Take the standard simplex $T_{10}=\operatorname{conv}(0,e_1,\ldots,e_{10})$ and the product $K=T_{10}\times T_{10}\subset\mathbb R^{20}$. The exact asserted comparison is \[ \frac{R_{20}(K)}{c_{20}}=\frac{121\binom{20}{10}}{21\cdot2^{20}} =\frac{22\,355\,476}{22\,020\,096}>1. \] Thus a simplex does not maximize projection-body volume normalized by $|K|^{d-1}$ in every dimension.

Outline of the argument

For manuscript 1 (family 088; 2026-09-24), facet-vector formulas give an exact projection-body volume for a simplex and its Cartesian products. Comparing the resulting finite quantities produces a ratio strictly above one; the product formula also explains the large-dimensional excess.

Formal verification

For manuscript 1 (family 088; 2026-09-24), OpenAI lists Lean statements for this assertion: taking the Cartesian product of two dimension-ten simplices yields a body whose normalized projection volume exceeds the simplex value in dimension twenty, refuting the proposed maximum.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are PettyProjectionVolume.lean, ProjectionCounterexample.lean, ProjectionVolume.lean.

Their named declarations include OAI.Paper092.product_counterexample, OAI.PettyProjection.petty_projection_volume, OAI.ProjectionCounterexample.universal_simplex_upper_bound_false.

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

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