Bourgain's slicing conjecture

For a convex body $K\subset\mathbb{R}^n$, define its covariance matrix by

$$\operatorname{Cov}_{ij}(K)=\int_K x_i x_j\frac{dx}{|K|}-\int_K x_i\frac{dx}{|K|}\int_K x_j\frac{dx}{|K|},$$

and define

$$\mathcal{C}(K)=\frac{|K|^2}{\det\operatorname{Cov}(K)}.$$

The isotropic constant $L_K$ satisfies $\mathcal{C}(K)=1/L_K^{2n}$.

Bourgain's slicing conjecture. There exists a constant $c>0$ independent of dimension such that

$$\mathcal{C}(K)\geq c^n$$

for all $n\in\mathbb{N}$ and all convex bodies $K\subset\mathbb{R}^n$. This is the slicing problem, equivalently a dimension-independent upper bound on the isotropic constant. The source presents it as Bourgain's conjecture and gives no resolution, so it remains open.