An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture (Q2035530)

A probability density function \(p\): \(\mathbb{R}^d \to \mathbb{R}\) is log-concave if its logarithm is concave. Common probability distributions such as normal, exponential and logistic are log-concave. The isoperimetric coefficient \(\psi(p)\) of a density \(p\) in \(\mathbb{R}^d\) is defined as \(\psi(p) = \inf\limits_{S\subset\mathbb{R}^d} \frac{p^+(\partial S)} {\min(p(S), p(S^c))}\), where \(S^c\) is the complement of \(S\), \(p(S) = \int_{x\in S} p(x)\,dx\), and the boundary measure of the subset is \(p^+(\partial S) = \liminf\limits_{\varepsilon\to 0^+} \frac{p(\{x: d(x, S)\le \varepsilon\}) - p(S)}{\varepsilon}\), where \(d(x, S)\) is the Euclidean distance between \(x\) and the subset \(S\). Kannan, Lovàsz and Simonovits (KLS) conjecture is stated as follows: There exists a universal constant \(c\), such that for any log-concave density \(p\) in \(\mathbb{R}^d\), we have \(\psi(p) \ge \frac{c}{\sqrt{\rho(p)}}\), where \(\rho(p)\) is the spectral norm of the covariance matrix of \(p\) [\textit{R. Kannan} et al., Discrete Comput. Geom. 13, No. 3--4, 541--559 (1995; Zbl 0824.52012)]. Proving the lower bound on \(\psi(p)\) up to some small factors is the main goal of the present paper. Additionally, improving the lower bound in the KLS conjecture also improves concentration inequalities for Lipschitz functions of log-concave measures. It also leads to faster mixing time bounds of Markov chain Monte Carlo (MCMC) sampling algorithms on log-concave measures.