The mixed discriminant of a family of point configurations can be considered as a generalization of the
A
A
-discriminant of one Laurent polynomial to a family of Laurent polynomials. Generalizing the concept of defectivity, a family of point configurations is called defective if the mixed discriminant is trivial. Using a recent criterion by Furukawa and Ito we give a necessary condition for defectivity of a family in the case that all point configurations are full-dimensional. This implies the conjecture by Cattani, Cueto, Dickenstein, Di Rocco, and Sturmfels that a family of
n
n
full-dimensional configurations in
Z
n
{\mathbb {Z}}^n
is defective if and only if the mixed volume of the convex hulls of its elements is
1
1
.