The Sharp Threshold for the Dyn-Farkhi Conjecture
Dyn and Farkhi conjectured that the squared Hausdorff distance from a compact set to its convex hull is subadditive under Minkowski addition. It holds in dimensions one and two and fails from dimension three; the sharp threshold exponent for Hausdorff convexification is now determined.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Convex geometry
- Posed by
- Nira Dyn, Elza Farkhi
- Year posed
- 2004
- Years open
- 22y
- Solved
- 2026-06-09
- Model
- ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Peter van Hintum
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
pins the sharp threshold; the conjecture itself was already known false above dimension two
What the AI did
The acknowledgement thanks ChatGPT 5.5 Pro for helping prove Theorem 1.4 and for generating the TikZ code for the figures, so the credit names a specific theorem rather than the paper as a whole.
Verification
Single-author arXiv preprint; not yet peer-reviewed.
Source
arXiv:2606.10815 - The sharp threshold for Hausdorff convexification under Minkowski addition