Minkowski's theorem says that if we have a nonsingular $n\times n$ matrix $A$, then any symmetric convex body in $R^n$ with volume more than $2^n \det(A)$ contains a lattice point of $A$.
What can we say if $A$ is a $n\times m$ matrix for $m>n$? Clearly we may take any submatrix of dimensions $n\times n$ and apply the theorem on it. My question is if we can do better than saying any convex symmetric with volume more than $2^n \min(\det(B))$ where $B$ runs over the submatrices of size $n\times n$ contains a lattice point.
We also note that there is a $0$ linear combination of the columns, but since we can't control coefficients I don't see how we may use this.