Characterizing D-optimal Rotatable Designs with Finite Reflection Groups

We establish a powerful construction of D-optimal Euclidean designs, or D-optimal rotatable designs, on the unit hyperball by using the corner vectors associated with the symmetry groups of (semi-)regular polytopes. This is a full generalization of the classical construction choosing points in the form (a,…,a,0,…,0) or in their orbits under the symmetry group of a regular hyperoctahedron (Gaffke and Heiligers 1995b), (Hirao et al. 2014), as well as Scheffé’s {n, 2}-lattice design on the simplex. We prove a Gaffke-Heiligers type theorem for D n - and A n -invariant D-optimal Euclidean designs which is a “reduction theorem” on the computational cost of searching observation points, and thereby construct many families of D-optimal Euclidean designs. For each group A n , D n , B n , H 3, H 4, F 4, E 6, E 7, E 8, we determine the maximum degree of a D-optimal Euclidean design constructed by our method and in particular discover examples of degrees 5 and 6 for E 8 and H 4, respectively. We also classify such maximum-degree designs for the groups H 3, H 4 and F 4 acting on the 3- and 4-dimensional Euclidean spaces.