Notions of obvious breadth and importance should, when possible, be examined under a number of different disciplinary lenses. This is the spirit in which the present offering by Julius Barbanel (a mathematician) joins recent books by Hervé Moulin (an economist) [32] and Nicholas Rescher (a philosopher) [35]. But fairness – or, more explicitly, fair division – comes in a number of different flavors, and we should begin by setting forth something of a general framework in which we can place the present book.

One of the more important dichotomies in the treatment of fairness is the extent to which the treatment is normative. Is the author trying to argue that certain methods of allocation are superior to others? The treatment of fair division by economists, philosophers, and political scientists tends to lie in the normative camp. Mathematicians, on the other hand, focus on what is possible and what is not, and often leave subjective judgments to others, as Barbanel does here.

Yet there is a normative aspect of the present work that sets it apart from the great majority of mathematical treatments, and it is revealed in Barbanel's choice of title. The work is not called “The Geometry of Fair Division” but “The Geometry of Efficient Fair Division.” Efficiency – also called Pareto optimality, after the nineteenth-century Italian scholar Vilfredo Pareto – is, according to Hervé Moulin, “the single most important tool of normative economics” [32, pg. 8].