Brannen's simplex conjecture for projection-body volume

Theorem 1: for $K=T_{10}\times T_{10}\subset\mathbb R^{20}$ , $R_{20}(K)/c_{20}=121\binom{20}{10}/(21\cdot2^{20})>1$ , so the simplex bound fails in dimension 20; iterating the product identity $R_{r+s}(A\times B)=R_r(A)R_s(B)$ gives $R_n(K_n)\ge\lambda^nc_n$ for some $\lambda>1$ and all large $n$ . Priority: Feng, Hu, Liu and Xu (preprint, August 2026) already disproved the conjecture in every dimension $d\ge9$ , and Chen et al. proved it in dimension 3; the paper cites both and presents its result as a direct product construction with an exact value. It does not determine the optimal upper constant or the maximizers.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Brannen's simplex conjecture for projection-body volume · Brannen's simplex conjecture

Confidence: Not scored

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VibeMathed record: Brannen's simplex conjecture for projection-body volume

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No independent mathematician has checked this yet. Checked here: Theorem 1 was read against Brannen's conjecture as the paper states it; it claims $R_{20}(T_{10}\times T_{10})/c_{20}=22355476/22020096>1$ . The proof (a product identity for projection bodies of polytopes plus an exact simplex computation) was not refereed. The Lean challenges ComparatorChallenges/ProjectionCounterexample.lean (declaration OAI.ProjectionCounterexample.universal_simplex_upper_bound_false, in lean/formalization.yaml) and ComparatorChallenges/ProjectionVolume.lean (OAI.Paper092.product_counterexample) were read here: the first asserts that not every convex body in $\mathbb R^{20}$ has ratio at most that of the 20-simplex, the second gives the exact value for the product of two 10-simplices. This states the disproof. Not rebuilt here. The paper's exponential-excess corollary is not formalized. The conjecture had already been refuted for every $d\ge9$ by Feng, Hu, Liu and Xu (August 2026), as the paper itself says.