Kannan-Lovasz-Simonovits Spectral Gap Conjecture

Definitions and mathematical statement

The supplied record concerns the Kannan--Lovász--Simonovits (KLS) conjecture and reports that Yuansi Chen presented an improved spectral-gap bound for convex sets and log-concave probability measures. It does not supply a mathematical formulation of that conjecture or an explicit spectral-gap inequality.

The qualitative description of the remaining dimension dependence can be made precise conditionally: for a positive dimension-dependent loss factor $D\colon{\mathbb{N}}_{\geq 1}\to(0,\infty)$, growth slower than every positive power means \[ \forall\varepsilon>0\;\exists C_\varepsilon\in(0,\infty)\;\forall n\in{\mathbb{N}}_{\geq 1},\qquad D(n)\leq C_\varepsilon n^\varepsilon. \] Here $n$ denotes dimension. This explains the growth terminology only; the record does not identify $D$, specify where it enters the spectral-gap bound, or state the normalization needed for a dimension-free conjecture.

Unresolved mathematical statement. No separate, complete conjecture can be recovered from this record. Its supplied statement status is unrecoverable. The supplied verification note identifies the passage as a workshop-level introduction that merely mentions KLS and says that a complete formulation is retained in the separate source record OWR-8415364-006. That formulation is not included here. Accordingly, neither a normalized KLS inequality nor an independent open problem is asserted in this draft.

Short English statement

This record preserves introductory workshop context about the Kannan--Lovász--Simonovits conjecture and Chen's reported improvement with subpolynomial dimension dependence. It contains no complete mathematical problem. The supplied dated curation labels it partially solved and describes the dimension-free conjecture as unresolved; these are supplied assessments requiring independent review.

Sources

• Local registry: permanent UnsolvedMath ID $30004943$, exact source problem code catalogue entry, title "Kannan-Lovasz-Simonovits Spectral Gap Conjecture," in the Oberwolfach Reports Open Problems collection. The supplied record labels the statement unrecoverable and the problem partially_solved; no proposer is specified, and the proposed-year field is 2021.
• Supplied report citation: Convex Geometry and its Applications (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/59, pp. 3187--3266. https://doi.org/10.4171/owr/2021/59. The record attributes the presentation of the improved bound to Yuansi Chen and names Kannan, Lovász, and Simonovits in connection with the conjecture.
• Supplied literature source, "Chen, almost constant KLS lower bound": https://doi.org/10.1007/s00039-021-00558-4.
• Supplied literature source, "Chen primary preprint": https://arxiv.org/abs/2011.13661.
• Supplied literature source, "current KLS status": https://www.solveall.org/problem/kls-conjecture.
• Supplied literature assessment, checked 2026-08-22: Chen proved the subpolynomial-dimensional lower bound highlighted in the record, while the dimension-free KLS conjecture remained unresolved as of that check date. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• Supplied statement-verification note: the official report's introductory passage merely mentions KLS, with the complete formulation retained in source record OWR-8415364-006; no separate clean statement is promoted for the present record. This is a supplied verification claim, not a claim that the report or the other record was consulted for this draft.