  An integer point in a polyhedron is called irreducible iff it is not the
midpoint of two other integer points in the polyhedron. We prove that the
number of irreducible integer points in $n$-dimensional polytope with radius
$k$ given by a system of $m$ linear inequalities is at most
$O(m^{\lfloor\frac{n}{2}\rfloor}\log^{n-1} k)$ if $n$ is fixed. Using this
result we prove the hypothesis asserting that the teaching dimension in the
class of threshold functions of $k$-valued logic in $n$ variables is
$\Theta(\log^{n-2} k)$ for any fixed $n\ge 2$.
