Minimal-volume projections of cubes and totally unimodular matrices (Q1870065)

The image of the unit cube \(K^m\subset\mathbb{R}^m\) under a linear projection from \(\mathbb{R}^m\) onto a fixed \(l\)-dimensional linear subspace \(L\) of \(\mathbb{R}^m\) is called a minimal-volume projection if it has minimal \(l\)-dimensional volume among all such projection images in \(L\). The following result is proved. An \(l\)-dimensional zonotope is linearly equivalent to a minimal-volume projection of \(K^m\) if and only if it is linearly equivalent to the zonotope spanned by the multiples of rows of a totally unimodular \(m\times r\) matrix of rank \(l\).