Optimal Constant Problem for n-cycle Solved
michaelchchoi · x · 2026-07-15
The author, collaborating with Rupert Frank, published a new paper solving the problem of the optimal constant in the log-Sobolev inequality on the n-cycle.
The core conclusion states that for all n ≥ 4, this optimal constant equals half of the spectral gap. The text also reviews the history of this problem:
- The issue originally dates back to Diaconis–Saloff-Coste (1995)
- Even n=2k was solved by Chen–Sheu (2003)
- n=5 was solved by Chen–Liu–Saloff-Coste (2008)
- Odd n ≤ 21 were previously covered by computer-assisted proofs from Faust–Fawzi (2021)
- n=3 remains an exceptional case
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