On the non-sufficiency of total unimodularity of the constraint matrix in the definition of an integer polytope

Question

_{Crossposted at Operations Research SE}

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Is there an example of an $m\times n$ integer matrix $A$ and an integer vector $b\in \mathbb {Z}^{m}$ such that the polyhedron $P := \{ x\in \mathbb {R}^{n} \mid A x \leq b\}$ is integer, while $A$ is not totally unimodular?

Note that a polyhedron is integer if all of its vertices are integral.

Answer 1

Yes, there is. One could choose

$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ -1 & -a\end{pmatrix}$ and $b = \begin{pmatrix} 0 \\ 0 \\ a\end{pmatrix}$

for $a$ being a positive integer.

The polytope $P$ is a simplex/triangle with vertices

$\begin{pmatrix} 0 \\ 0\end{pmatrix}$, $\begin{pmatrix} 0 \\ -1\end{pmatrix}$ and $\begin{pmatrix} -a \\ 0\end{pmatrix}$

and the maximal minor in absolute value of $A$ is $a$ which can be arbitrary large.