Discrete slicing problems

The well-known  slicing problem in Convex Geometry asks whether
there exists an absolute constant $c$ so that for every
origin-symmetric convex body $K$  of volume 1 there is a hyperplane
section of $K$ whose $(n − 1)$-dimensional volume is greater than $c$.
Motivated by a question  of Alexander Koldobsky, we are studying
a similar  slicing problem (and related problems)
when the volume functional is  replaced by the lattice point enumerator.

(Based on joint works with Matthew Alexander, Sören L. Berg and Artem Zvavitch)