Hyperplane conjecture (bounded isotropic constant K_d)

Establish whether the maximal isotropic constant K_d, defined as the supremum of M(X)^{2/d}·σ over all isotropic log-concave probability distributions X on R^d with covariance matrix Cov(X)=σ^2 I_d and with M(X) the essential supremum of the density of X, is bounded above by a universal constant independent of the dimension d.

Background

The paper discusses bounds on the maximum of a density M(X) in relation to the covariance matrix of a random vector, introducing the isotropic constant M(X)^{2/d}·σ as a key quantity. Within the class of isotropic log-concave distributions, the authors note that lower bounds on this quantity can be reversed to yield an upper bound of the form M^{2/d}·σ ≤ K_d.

They then recall the hyperplane (slicing) conjecture (attributed to Bourgain), which asserts that K_d is bounded by an absolute constant independent of d. The conjecture is known to hold under additional symmetry assumptions (e.g., unconditional symmetry), while in the general log-concave case the best current bound is due to Klartag: K_d^2 ≤ C·log(d+1).

This conjecture is central in asymptotic geometric analysis and directly relevant to the paper’s theme because it controls the isotropic constant that appears in several of their bounds, including those for characteristic function separation and Berry–Esseen-type local limit estimates.