Optimal constants in free KLS and thin-shell inequalities

Determine the optimal universal constants and extremal behavior in the free KLS inequality and the associated free thin-shell estimate under CD(0,∞).

Background

The paper proves a dimension-free free KLS bound with constant 10 and derives a corresponding free thin-shell estimate. It explicitly leaves unresolved whether these constants are optimal and what the extremal examples are, noting parallels with the classical theory.

References

We do not address the optimal constants in the free KLS and thin-shell inequalities in this paper. Various computations nevertheless suggest intriguing parallels between the classical and free theories, even at the level of sharp constants and extremal behaviour. We leave these questions for further investigation.

— Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound  (2610.22109 - Diez, 15 Oct 2026) in Remark following Corollary \ref{cor:free-thin-shell}, Section 3, subsection “A free KLS bound”