Fair cake-cutting

Fair cake-cutting is a kind of fair division problem. The problem involves a heterogeneous resource, such as a cake with different toppings, that is assumed to be divisible – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be subjectively fair, in that each person should receive a piece that he or she believes to be a fair share.

The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time.

The cake-cutting problem was introduced by Hugo Steinhaus after World War II and is still the subject of intense research in mathematics, computer science, economics and political science.

There is a cake C, which is usually assumed to be either a finite 1-dimensional segment, a 2-dimensional polygon or a finite subset of the multidimensional Euclidean plane R^d.

There are n people with equal rights to C.

C has to be divided to n disjoint subsets, such that each person receives a disjoint subset. The piece allocated to person i is called ${\displaystyle X_{i}}$, and ${\displaystyle C=X_{1}\sqcup \cdots \sqcup X_{n}}$.

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