This book develops a new theory in convex geometry generalizing positive bases and related to Carathordorys Theorem by combining convex geometry the combinatorics of infinite subsets of lattice points and the arithmetic of transfer Krull monoids the latter broadly generalizing the ubiquitous class of Krull domains in commutative algebraThis new theory is developed in a selfcontained way with the main motivation of its later applications regarding factorization. While factorization into irreducibles called atoms generally fails to be unique there are various measures of how badly this can fail. Among the most important is the elasticity which measures the ratio between the maximum and minimum number of atoms in any factorization.