Φp-optimal second order designs for symmetric regions

Designs for quadratic regression on a cube, on a cube with truncated vertices and on a ball are studied in terms of a family of criteria, introduced by Kiefer (1974, 1975), that includes A-, D- and E-optimality. Both theoretical and numerical results on structure and performance are presented. In particular, D- and E-optimal designs are described and a procedure of construction of nearly robust (under variation of criterion) integer designs is suggested. Some examples are given for dimensions 4, 5 and 6.