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Problem 1040
Let be a closed infinite set, and let be the infimum of
as ranges over all polynomials of the shape with .
Is determined by the transfinite diameter of ? In particular, is whenever the transfinite diameter of is ?
Source: erdosproblems.com/1040
- Analysis
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Claimed, AnsweredA full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.
Open. The site's commentary (last edited 1 February 2026) credits Aletheia [Fe26] with the negative answer to the first question, and the proof-claims tab carries three full proof claims on the second question, with AI systems named on the tab: by shlummi and by Declan Gessel (6 September 2026) and by Ioannis Tzachristas (9 September 2026); the site labels the problem OPEN. The accepted partial claims are the segment and disc cases of Erdős, Herzog and Piranian, the regular capacity-above-one case of Krishnapur, Lundberg and Ramachandran and Ghosh and Ramachandran's counterexamples and capacity-above-one case; the pending partial claims are Aletheia's capacity-zero pair, the smooth capacity-one case of Krishnapur, Lundberg and Ramachandran, Tzachristas's page, shlummi's page, Gessel's page and Kitamura's page.