Integer proportional sharing problems mainly arise in electoral studies: apportionment of seats among states, or repartition of seats among the candidate lists in a ballot. This problem of fair sharing has given rise to various mathematical studies. These studies, culminating in Balinski and Young’s Fair Representation, Meeting the Ideal of One Man, One Vote [Yale University Press, New Haven, CT, 1982], fail to yield a global and satisfactory theory of proportional representation because they almost exclusively focus on a particular family of “multiplicative processes.” In order to build such a theory, several new concepts are introduced and two different approaches are considered to the problem, an algorithmic one and a geometric one. Both approaches prove the existence of a family of additive sharing processes in parallel to multiplicative ones, and the geometric approach allows both families to merge into a new wide family of processes containing all classical methods as particular cases. A series of p...