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on KLS constant for log-concave probability measures ($C_{20c}$)

The statement

$C_{20c}$ is the KLS constant (Kannan–Lovász–Simonovits constant) for log-concave measures. It is defined as

C_{20c} := \sup_{n\ge 1} \psi_n,

where $\psi_n$ is the worst-case inverse Cheeger (isoperimetric) constant among isotropic log-concave probability measures on $\mathbb R^n$ .

More precisely, let $\mu$ be a log-concave probability measure on $\mathbb R^n$ (i.e. $\mu$ has density $\rho(x)=e^{-V(x)}$ for some convex $V:\mathbb R^n\to\mathbb R\cup\{+\infty\}$ ). For a Borel set $A\subset\mathbb R^n$ , define the (outer) Minkowski boundary measure

\mu^+(A) := \liminf_{\varepsilon\to 0^+} \frac{\mu(A_\varepsilon)-\mu(A)}{\varepsilon}, \qquad A_\varepsilon := \{x\in\mathbb R^n:\operatorname{dist}(x,A)\le \varepsilon\}.

The Cheeger constant (isoperimetric coefficient) of $\mu$ is

h_\mu := \inf_A \frac{\mu^+(A)}{\min(\mu(A),1-\mu(A))},

and the corresponding inverse Cheeger constant is

\psi_\mu := \frac{1}{h_\mu}.

We say that $\mu$ is isotropic if it has barycenter $0$ and covariance matrix $\mathrm{Cov}(\mu)=I_n$ . One then defines

\psi_n := \sup\{\psi_\mu : \mu \text{ is an isotropic log-concave probability measure on }\mathbb R^n\}.

The KLS conjecture asserts that $C_{20c}<\infty$ , i.e. that $\psi_n=O(1)$ uniformly in $n$ (and, in a stronger form, that the infimum defining $h_\mu$ is attained up to constants by half-spaces).

It is often convenient to work with the Poincaré (spectral gap) constant $C_P(\mu)$ , defined as the smallest constant such that

\mathrm{Var}_\mu(f)\le C_P(\mu)\int |\nabla f|^2\,d\mu

for all smooth enough $f$ . For log-concave measures, $C_P(\mu)$ is equivalent up to universal factors to $\psi_\mu^2$ ; for instance one has

\frac{1}{\pi}\,\psi_\mu^2 \ \le\ C_P(\mu)\ \le\ 4\,\psi_\mu^2.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.