  In 1921 Blichfeldt gave an upper bound on the number of integral points
contained in a convex body in terms of the volume of the body. More precisely,
he showed that $#(K\cap\Z^n)\leq n! \vol(K)+n$, whenever $K\subset\R^n$ is a
convex body containing $n+1$ affinely independent integral points. Here we
prove an analogous inequality with respect to the surface area $\F(K)$, namely
$ #(K\cap\Z^n) < \vol(K) + ((\sqrt{n}+1)/2) (n-1)! \F(K)$. The proof is based
on a slight improvement of Blichfeldt's bound in the case when $K$ is a
non-lattice translate of a lattice polytope, i.e., $K=t+P$, where
$t\in\R^n\setminus\Z^n$ and $P$ is an $n$-dimensional polytope with integral
vertices. Then we have $#((t+P)\cap\Z^n)\leq n! \vol(P)$.
  Moreover, in the 3-dimensional case we prove a stronger inequality, namely
$#(K\cap\Z^n) < \vol(K) + 2 \F(K)$.
