Waldschmidt constants of symmetric sets of points in $\mathbb{P}^3$

Configurations of points defined by complex reflection groups have attracted a lot of attention recently in several directions of research, e.g., the containment problem between ordinary and symbolic powers of ideals, in the theory of unexpected hypersurfaces, in the study of sets of points whose general projections are complete intersections and in the ideas revolving around the Bounded Negativity Conjecture. In order to understand these configurations better several attempts have been undertaken to compute their various invariants. In this paper we focus on their Waldschmidt constants. In the case of plane configurations of points determined by reflection groups most (but not all) Waldschmidt constants are known. Here we pass to configurations in $\mathbb{P}^3$ where new ideas are required as the identification between divisors and curves is no longer available. In particular, we precisely compute the Waldschmidt constant for configurations of points in $\mathbb{P}^3$ coming from the $D_4,B_4,F_4$, and $H_4$ root systems.