Abstract: | We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space n. For each i=1, . . . , m, there is a class of subsets Mij of Tn. Assume that for each i=1, . . . , m, that Mij is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component xjB(i, j) imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem. |