  The mixed discriminant of a family of point configurations can be considered
as a generalization of the $A$-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 $n$ full-dimensional
configurations in $\mathbb{Z}^n$ is defective if and only if the mixed volume
of the convex hulls of its elements is $1$.
