Kannan–Lovasz–Simonovits (KLS) isoperimetric conjecture
Prove that there exists a universal constant C > 0 such that for every log-concave probability measure μ on R^n, the isoperimetric constant satisfies ψ_μ ≥ C inf_{halfspaces H} μ^+(H)/min{μ(H), μ(R^n \ H)}, equivalently ψ_μ ≥ C / √(||A||_{op}), where A is the covariance of μ; in particular, establish ψ_n bounded below by a universal constant for isotropic log-concave measures.
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Indeed, Kannan, Lov\u00e1sz and Simonovits conjectured in 1995 that for all log-concave measures, the isoperimetric constant is, up to a universal constant, attained by minimizing over half-spaces, like in the Gaussian case: There exists a universal constant $C>0$ such that for every log-concave measure $\mu$, the isoperimetric constant satisfies \begin{align*} \psi_\mu \geq C \inf_{H \subseteq \mathbb{R}n, H \text{ is halfspace} \frac{\mu+(S)}{\min{\mu(S), \mu(\mathbb{R}n\backslash S)}. \end{align*} Equivalently, there exists a universal constant $C>0$ such that \begin{align*} \psi_\mu \geq \frac{C}{\sqrt{|A|_\text{op}, \end{align*} where $A$ denotes the covariance matrix of the measure $\mu$, $A=\mathbb{E} \left(X- \mathbb{E}X\right)\left(X- \mathbb{E}X\right)\top$, where $E$ denotes expectation of $X\sim\mu$.
An isoperimetric conjecture of Kannan--Lovasz--Simonovits (KLS) asserts that for any convex body $K \subset Rn$, one has: \begin{equation} \label{eq:intro-KLS} D_{Che}(K)2 \geq \frac{c}{Cov_K}{op} , \end{equation} where $c > 0$ is a dimension-independent constant (and the Cheeger constant $D{Che}(K)$ is defined analogously as for $T_{\Lambda}$).