  We investigate $(0,1)$-matrices that are {\em convex}, which means that the
ones are consecutive in every row and column. These matrices occur in discrete
tomography. The notion of ranked essential sets, known for permutation
matrices, is extended to convex sets. We show a number of results for the class
$\mc{C}(R,S)$ of convex matrices with given row and column sum vectors $R$ and
$S$. Also, it is shown that the ranked essential set uniquely determines a
matrix in $\mc{C}(R,S)$.
