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Math-2633-A1 · revision r1 · on Math-2633

OpenAI claim summary for Math-2633

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Simplex maximum for projection-body volume
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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