Minimal Terracini loci in projective spaces

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.