Title: Characterizing D-optimal Rotatable Designs with Finite Reflection Groups

Author(s): Masanori Sawa and Masatake Hirao
Issue: Volume 79 Series A Part 1 Year 2017
Pages: 101 -- 132
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 $(a, \ldots, a, 0, \ldots, 0)$ or in their orbits under the symmetry group of a regular hyperoctahedron (Gaffke and Heiligers 1995b), (Hirao et al. 2014), as well as Scheffe’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.
AMS (2000) subject classification . 62K05, 65D32, 05E99.
Keywords and phrases: Approximate design, Optimal design, Euclidean design, Corner-vector method, D-optimality, Finite irreducible reflection group, Gaffke-Heiligers theorem, Invariant harmonic polynomial