The slicing problem for sections of proportional dimensions

The slicing problem asks whether there exists an absolute constant $c$
such that for every dimension $n$ and every symmetric convex body $K$ of
volume one in $R^n$ there exists a central hyperplane section of $K$
with $(n-1)$-dimensional volume greater than $c$. The problem is still
open, and the best known result $c\sim n^{-1/4}$ is due to Klartag, who
removed a logarithmic term from an earlier estimate of Bourgain. In this
talk we prove a lower dimensional version of the slicing problem with
sections of dimension $\lambda n,$ where $\lambda\in (0,1).$ The
constant depends only on $\lambda.$ We also consider a generalization of
the slicing problem to arbitrary measures in place of volume.