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

On lattice width of lattice-free polyhedra and height of Hilbert bases