Polynomial Bounds in Koldobsky's Discrete Slicing Problem

In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$ dimensional linear subspace $H\subset\mathbb{R}^n$ with $$|K\cap\mathbb{Z}^n| \leq d_n\,|K\cap H\cap \mathbb{Z}^n|\,\mathrm{vol}(K)^{\frac 1n}.$$ In this article we show that $d_n$ is bounded from above by $c\,n^2\,\omega(n)$, where $c$ is an absolute constant and $\omega(n)$ is the flatness constant. Due to the best known upper bound on $\omega(n)$ this gives a ${c\,n^{10/3}\log(n)^a}$ bound on $d_n$ where $a$ is another absolute constant. This bound improves on former bounds which were exponential in the dimension

Bounds on the lattice point enumerator via slices and projections

Gardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.TU Berlin, Open-Access-Mittel – 202

Interpolating between volume and lattice point enumerator with successive minima

We study inequalities that simultaneously relate the number of lattice points, the volume and the successive minima of a convex body to one another. One main ingredient in order to establish these relations is Blaschke's shaking procedure, by which the problem can be reduced from arbitrary convex bodies to anti-blocking bodies. As a consequence of our results, we obtain an upper bound on the lattice point enumerator in terms of the successive minima, which is equivalent to Minkowski's upper bound on the volume in terms of the successive minima