Market allocation of indivisible goods

The existence of equitable allocations of divisible goods is established. The methods used give divisions of a good into geometrically simple sets, such as simplexes or polyhedral convex cones. This economic problem is solved by applying David Gale's covering lemma. An extension of Gale's lemma and its dual versions are established; the proofs are based on Ky Fan's fundamental theorem on coincidence of two set-valued functions.