  In this work we focus on saturated $D$-optimal designs. Using recent results,
we identify $D$-optimal designs with the solutions of an optimization problem
with linear constraints. We introduce new objective functions based on the
geometric structure of the design and we compare them with the classical
$D$-efficiency criterion. We perform a simulation study. In all the test cases
we observe that designs with high values of $D$-efficiency have also high
values of the new objective functions.
