  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 $O(d)^{d-m}$.
