Georgia Institute of Technology College of Sciences

In this talk we will discuss an answer to a question of Alexander Koldobsky and present a discrete version of his slicing inequality. We let $\# K$ be a number of integer lattice points contained in a set $K$. We show that for each $d\in \mathbb{N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K \subset \mathbb{R}^d$ containing $d$ linearly independent lattice points $$ \# K \leq C(d)\text{max}_{\xi \in S^{d-1}}(\# (K\cap \xi^\perp))\, \text{vol}_d(K)^{\frac{1}{d}},$$where $\xi^\perp$ is the hyperplane orthogonal to a unit vector $\xi$ .We show that $C(d)$ can be chosen asymptotically of order $O(1)^d$ for hyperplane slices. Additionally, we will discuss some special cases and generalizations for this inequality. This is a joint work with Martin Henk and Artem Zvavitch.