Minimal matrices and discrete tomography

In this paper we deal with the question of when an n-dimensional (0, 1)-matrix is uniquely determined by its 1-marginals and with the related notion of additivity. We present a survey of known results; several of them have been considered before only in dimensions 2 and 3. Here, we show how to extend them to any dimension. The main results are characterizations of uniqueness and additivity: one of algebraic nature, involves matrices with integer entries; other of geometric nature, uses transportation polytopes and permutohedra.