Submajorization and the Geometry of Unordered Collections

In this paper, we use submajorization to compare distances between either multisets of real numbers or multisets modulo translation on the real line. We provide a geometrical interpretation in which multisets are represented by points in an orbifold and bijections between multisets are represented by paths between these points. This interpretation shows that submajorization is closely related to the geometrical principle that the shortest path between two points is a straight line. Our results have applications to diverse problems from economics to music theory; moreover, they suggest generalizations of statistical measures of the center and spread of a distribution.