In 1988, Mathieu Meyer presented a lower bound on the volume of a convex body in terms of the volumes of its sections with the coordinate hyperplanes.
Our aim is to "discretize" this inequality by replacing the volume by the lattice point enumerator. It turns out that such a discrete analog cannot exist for general convex bodies. Moreover, it will not imply its continuous counterpart.
We prove inequalities for the case of o-symmetric bodies (using arithmetic progressions) and unconditional bodies (by proving a universal bound for the lattice point enumerator of unconditional lattice polytopes). Moreover, the talk will touch on a reverse inequality and other related problems.