ABSTRACT
The problem of dividing an object among two or more parties, known as the fair division problem has captured the attention of several notable mathematicians, such as Steinhaus, Banach, Knaster, Dubins and Spanier. The latter two authors, in particular, have defined two different optimization problems that single out satisfactory partitions of the object in question.
In this work we review the available results on optimization in fair division and we take a closer look at some open questions. In particular we add constraints to the first problem, in order to improve the quality of the optimal solution. We recur to functional Lagrangian duality in order to solve the constrained problem. Furthermore, we describe a constructive procedure (otherwise unavailable in the existing literature) to find a solution to the second problem, based on a geometrical setting. We apply the algorithms to several examples.
