Let $\# K$ be a number of integer lattice points contained in a set $K$. In this paper we prove 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 $$ \# K \leq C(d)\max(\# (K\cap H))\, { \rm vol}_d(K)^{ \frac{ d-m}{ d}}, $$ where the maximum is taken over all $m$-dimensional subspaces of ${ \mathbb R}^d$. We also prove that $C(d)$ can be chosen asymptotically of order $O(1)^{ d}d^{ d-m}$. In addition, we show that if $K$ is an unconditional convex body then $C(d)$ can be chosen asymptotically of order $O(d)^{ d-m}$.
Categories: math.MG