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 minimal closed set containing V such that its complement is the 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 & lowast; such that these subsets collapse into a point O & lowast;, in the interior of T, is investigated; in such a case O & lowast; belongs to four spheres of radius R & lowast;, each one through three vertices of T and not containing the fourth one. As a consequence, the range of rho such that Vis a rho-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.