  We characterize the number of points for which there exist non-empty
Terracini sets of points in $\mathbb{P}^n$. Then we study minimally Terracini
finite sets of points in $\mathbb{P}^n$ and we obtain a complete description in
the case of $\mathbb{P}^3$, when the number of points is less than twice the
degree of the linear system.
