We give a characterization of all simplices of probability measures on a Cantor space X which may be realized as the simplex of all invariant probability measures for some minimal homeomorphism g of X. This extends a result of Akin for the case when K is a singleton, and an unpublished result of Dahl when K is finite-dimensional. All relevant notions of topological dynamics will be recalled. (Joint work with Tomás Ibarlucia)