  We consider the problem of computing a sequence of rankings that maximizes
consumer-side utility while minimizing producer-side individual unfairness of
exposure. While prior work has addressed this problem using linear or quadratic
programs on bistochastic matrices, such approaches, relying on Birkhoff-von
Neumann (BvN) decompositions, are too slow to be implemented at large scale.
  In this paper we introduce a geometrical object, a polytope that we call
expohedron, whose points represent all achievable exposures of items for a
Position Based Model (PBM). We exhibit some of its properties and lay out a
Carath\'eodory decomposition algorithm with complexity $O(n^2\log(n))$ able to
express any point inside the expohedron as a convex sum of at most $n$
vertices, where $n$ is the number of items to rank. Such a decomposition makes
it possible to express any feasible target exposure as a distribution over at
most $n$ rankings. Furthermore we show that we can use this polytope to recover
the whole Pareto frontier of the multi-objective fairness-utility optimization
problem, using a simple geometrical procedure with complexity $O(n^2\log(n))$.
Our approach compares favorably to linear or quadratic programming baselines in
terms of algorithmic complexity and empirical runtime and is applicable to any
merit that is a non-decreasing function of item relevance. Furthermore our
solution can be expressed as a distribution over only $n$ permutations, instead
of the $(n-1)^2 + 1$ achieved with BvN decompositions. We perform experiments
on synthetic and real-world datasets, confirming our theoretical results.
