R-hulloid of the vertices of a tetrahedron

The R -hulloid, in the Euclidean space R 3 , 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 ⁎ such that these subsets collapse into a point O ⁎ , in the interior of T , is investigated; in such a case O ⁎ 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 dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.