Correspondences between convex geometry and complex geometry

We present several analogies between convex geometry and the theory ofholomorphic line bundles on smooth projective varieties or K\"ahler manifolds.We study the relation between positive products and mixed volumes. We defineand study a Blaschke addition for divisor classes and mixed divisor classes,and prove new geometric inequalities for divisor classes. We also reinterpretseveral classical convex geometry results in the context of algebraic geometry:the Alexandrov body construction is the convex geometry version of divisorialZariski decomposition; Minkowski's existence theorem is the convex geometryversion of the duality between the pseudo-effective cone of divisors and themovable cone of curves.Comment: EpiGA Volume 1 (2017), Article Nr.