Optimal Partial Plank Coverings
Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the paper also settles every planar convex body.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Discrete Geometry, Coverings
- Posed by
- Karoly Bezdek
- Year posed
- —
- Years open
- —
- Solved
- 2026-07-29
- Model
- ChatGPT
- Vendor
- OpenAI
- Collaborators
- Egor Bakaev, Alexander Polyanskii
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper credits iterative discussions between the authors and ChatGPT with the proof of concavity of Minkowski subtraction - explicitly within the authors' own existing Minkowski-subtraction approach - and with the idea for the planar case. The overall approach is the authors'.
Verification
No independent check, and the model's contribution is bounded by the authors to two steps inside their own framework. Preprint, not refereed.