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

On minimally Terracini finite sets of points in projective spaces