  We present several analogies between convex geometry and the theory of
holomorphic line bundles on smooth projective varieties or K\"ahler manifolds.
We study the relation between positive products and mixed volumes. We define
and study a Blaschke addition for divisor classes and mixed divisor classes,
and prove new geometric inequalities for divisor classes. We also reinterpret
several classical convex geometry results in the context of algebraic geometry:
the Alexandrov body construction is the convex geometry version of divisorial
Zariski decomposition; Minkowski's existence theorem is the convex geometry
version of the duality between the pseudo-effective cone of divisors and the
movable cone of curves.
