Kannan–Lovász–Simonovits conjecture

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$. Write $\operatorname{Cov}(\mu)$ for its covariance matrix, $\|\operatorname{Cov}(\mu)\|_{op}$ for its operator norm, and $C_P(\mu)$ for its Poincaré constant.

Kannan–Lovász–Simonovits conjecture. There is a universal constant $C>0$ such that

$$\|\operatorname{Cov}(\mu)\|_{op} \leq C_P(\mu) \leq C\,\|\operatorname{Cov}(\mu)\|_{op}.$$

This conjecture asks whether the Poincaré constant of every log-concave measure is controlled, up to universal factors, by the largest variance of a linear function. It is a central problem concerning bottlenecks and functional inequalities in high-dimensional convexity; its resolution status is not specified in the source.