Forall-exist statements in pseudopolynomial time

Given a convex set $Q\subseteq R^m$ and an integer matrix $W\in Z^{m\times n}$, we consider statements of the form $\mathrm{\forall }b\in Q\cap Z^m$ $\mathrm{\exists }x\in Z^n$ s.t. $Wx\le b$. Such statements can be verified in polynomial time with the algorithm of Kannan and its improvements if $n$ is fixed and $Q$ is a polyhedron. The running time of the best-known algorithms is doubly exponential in~$n$. In this paper, we provide a pseudopolynomial-time algorithm if $m$ is fixed. Its running time is $(m\mathrm{\Delta }{)}^{O(m^2)}$, where $\mathrm{\Delta }=\mathrm{\parallel }W{\mathrm{\parallel }}_{\mathrm{\infty }}$. Furthermore it applies to general convex sets $Q$.