  $E$-optimal experimental designs for a second-order response surface model
with $k\geq1$ predictors are investigated. If the design space is the
$k$-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1
(1977a) 121-132] determined optimal designs in a restricted class of designs
(defined by the multiplicity of the minimal eigenvalue) and stated their
universal optimality as a conjecture. In this paper, we prove this claim and
show that these designs are in fact $E$-optimal in the class of all approximate
designs. Moreover, if the design space is the unit ball, $E$-optimal designs
have not been found so far and we also provide a complete solution to this
optimal design problem. The main difficulty in the construction of $E$-optimal
designs for the second-order response surface model consists in the fact that
for the multiplicity of the minimum eigenvalue of the "optimal information
matrix" is larger than one (in contrast to the case $k=1$) and as a consequence
the corresponding optimality criterion is not differentiable at the optimal
solution. These difficulties are solved by considering nonlinear Chebyshev
approximation problems, which arise from a corresponding equivalence theorem.
The extremal polynomials which solve these Chebyshev problems are constructed
explicitly leading to a complete solution of the corresponding $E$-optimal
design problems.
