Courtade's conjecture on volumes of Minkowski sums with the ball
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.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Zonoids
- Posed by
- Thomas Courtade
- Year posed
- 2017
- Years open
- 9y
- Solved
- 2026-08-13
- Model
- GPT-5.6 Sol
- Vendor
- OpenAI
- Collaborators
- Matthieu Fradelizi, Alfredo Hubard, Auttawich Manui, Cheikh Saliou Ndiaye, Shouda Wang, Artem Zvavitch
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Courtade conjectured that for convex bodies ,
The paper proves that this inequality is false in every dimension . In dimension , it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension , including a six-generator zonotope in dimension and smooth perturbative constructions in higher dimensions.
What the AI did
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.
Verification
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.
Sources
Submitted by VibeGene on