Courtade's conjecture on volumes of Minkowski sums with the ball

Precise statement

Although Courtade’s conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.6 Sol

The authors state that GPT-5.6 Sol was used during preparation of the paper as an auxiliary mathematical tool to explore examples, test determinant computations, and assist with preliminary proof development. The authors state that the final mathematical arguments, statements, and computations are theirs and that they independently verified them.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Unreviewed. arXiv 2608.12681 read here; the section "Acknowledgments and AI assistance disclosure" states that GPT-5.6 Sol was used as an auxiliary mathematical tool to explore examples, test determinant computations and assist preliminary proof development, and that the authors independently verified all arguments. Author-checked; not refereed; no formalization.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-13

   arXiv

   Courtade conjectured that for convex bodies $B,C\subset\mathbb R^n$, $$ (|B||C|)^{1/n} + (|B_2^n||B_2^n+B+C|)^{1/n} \le |B_2^n+B|^{1/n}|B_2^n+C|^{1/n}. $$ The paper proves that this inequality is false in every dimension $n\ge3$. In dimension $3$, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension $n\ge3$, including a six-generator zonotope in dimension $3$ and smooth perturbative constructions in higher dimensions.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.