The KLS (Kannan–Lovász–Simonovits) constant for log-concave measures

Upper bound, last-listed row

| $\psi_n \le C\sqrt{\log n}$ | [K2023] | Current best general bound (Theorem 1.2 of [K2023]). |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $\sqrt{\pi/2} \approx 1.25331$ | Classical | For the standard Gaussian measure, isoperimetric minimizers are half-spaces and $h_\gamma=\sqrt{2/\pi}$, hence $\psi_\gamma=\sqrt{\pi/2}$. |tracked since 2026-07-24 — no movement seen yet

Checked against material for its cited source

• Another row, not one shown above: bound row $\psi_n \le C(\log n)^5$ line 62 citing [KL2022] — the claim we checked: that the cited 2022 paper proves the inverse Cheeger constant of isotropic log-concave measures in dimension n is at most a constant times (log n) to the power 5, and calls it the first polylogarithmic bound, as the row's comment says. What we read could not settle it: version 2 of the preprint proves that the inverse Cheeger constant of isotropic log-concave measures is at most a constant times (log n) to the power 5, but version 1 gives the exponent as 5 + o(1), the journal edition the entry cites was not read, and the paper does not call its bound the first polylogarithmic one in those words. source link (read in arXiv version 2 (10 Apr 2022), printed pp. 2–3 as page images and the whole paper as text; version 1 (29 Mar 2022) as text around Theorem 1.1 only. The journal edition the entry cites, Geom. Funct. Anal. 32 (2022), pp. 1134–1159, was not reached: the publisher’s pages led to a sign-in). About the reference itself, not the bound: Crossref's record for the journal article and arXiv's abstract page for version 2 of the preprint both give the second author as Joseph Lehec; the table's reference entry for [KL2022] gives the name as “Jean Lehec”. Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.