R-hulloid of the vertices of a tetrahedron

The R-hulloid, in the Euclidean space R³, of the set of vertices V of a tetrahedron T is the mimimal closed set containing V such that its complement is union of open balls of radius R. When R is greater than the circumradius of T, the boundary of the R-hulloid consists of V and possibly of four spherical subsets of well defined spheres of radius R through the vertices of T. The existence of a value R^* such that these subsets collapse into one point O^* ∈ V is investigated; in such case O^* is in the interior of T and belongs to four spheres of radius R^*, each one through three vertices of T and not containing the fourth one. As a consequence, the range of ρ such that V is a ρ-body is described completely. This work generalizes to three dimensions previous results, proved in the planar case and related to the three circles Johnson's Theorem.