Primitive zonotopes arise naturally in various research areas, such as discrete geometry, combinatorial optimization, and theoretical physics. We provide geometric and combinatorial properties for these polytopes that allow us to estimate the size of their vertex sets. In particular, we show that the logarithm of the complexity of convex matroid optimization is quadratic, and we improve the bounds on the number of generalized retarded functions from quantum field theory. We also give a sharp asymptotic estimate for the number of vertices of a primitive zonotope that, in terms of Minkowski sums, is an intermediate between the permutohedra of types A and B.
