Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients

For mixture models an the simplex, we discuss the improvement of a given design in terms of increasing symmetry, as well as obtaining a larger moment matrix under the Loewner ordering. The two criteria together define the Kiefer design ordering. For the second-degree mixture model, we show that the set of weighted centroid designs constitutes a convex complete class for the Kiefer ordering. For four ingredients, the class is minimal complete. Of essential importance for the derivation is a certain moment polytope, which is studied in detail.