  We prove a sharp upper bound on the number of columns of a totally unimodular
matrix with column sums $1$ which improves upon Heller's bound by a factor of
$2$. The proof uses Seymour's decomposition theorem. Unimodular polytopes are
lattice polytopes such that the vertices of every full-dimensional subsimplex
form an affine lattice basis. This is a subclass of 0/1-polytopes and contains
for instance edge polytopes of bipartite graphs. Our main result implies a
sharp upper bound on the number of vertices of unimodular polytopes.
