Erdős-Herzog-Piranian Distance Products: Improved Lower Bound
Erdős, Herzog and Piranian (1958) asked whether the regular n-gon maximizes the product of pairwise distances among n points of fixed diameter. After the recent discovery that it does not for even n, this paper proves the first exponential improvement over the n-gon's value, via a vector-field technique.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Combinatorial geometry
- Posed by
- Paul Erdős, Fritz Herzog, George Piranian
- Year posed
- 1958
- Years open
- 67y
- Solved
- 2025-12-16
- Model
- ChatGPT
- Vendor
- OpenAI
- Collaborators
- Nat Sothanaphan
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 12 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
An improved lower bound on the maximal product; the sharp maximizer for the 1958 question remains unknown.
What the AI did
The acknowledgments thank "Stijn Cambie, ChatGPT, and Quanyu Tang for discussion (in alphabetical order)" - a discussion credit, listed alongside the human colleagues, with no specific step attributed.