  We study the lattice width of lattice-free polyhedra given by
$\mathbf{A}\mathbf{x}\leq\mathbf{b}$ in terms of $\Delta(\mathbf{A})$, the
maximal $n\times n$ minor in absolute value of
$\mathbf{A}\in\mathbb{Z}^{m\times n}$. Our main contribution is to link the
lattice width of lattice-free polyhedra to the height of Hilbert bases and to
the diameter of finite abelian groups. This leads to a bound on the lattice
width of lattice-free pyramids which solely depends on $\Delta(\mathbf{A})$
provided a conjecture regarding the height of Hilbert bases holds. Further, we
exploit a combination of techniques to obtain novel bounds on the lattice width
of simplices. A second part of the paper is devoted to a study of the above
mentioned Hilbert basis conjecture. We give a complete characterization of the
Hilbert basis if $\Delta(\mathbf{A}) = 2$ which implies the conjecture in that
case and prove its validity for simplicial cones.
