Face non-intersection property for FM-tetrahedra

Prove that for every FM-tetrahedron arising in the additively weighted Delaunay decomposition of a saturated packing of spheres with radii 1 and r = √2 − 1 in three-dimensional Euclidean space, no sphere centered at a vertex intersects the triangular face formed by the centers of the other three spheres (i.e., each sphere lies entirely on one side of the plane of the opposite face).

Background

FM-tetrahedra are defined as tetrahedra that may appear in the FM-decomposition (an additively weighted Delaunay triangulation) of a saturated packing of spheres of radii 1 and r = √2 − 1. For FM-triangles, two equivalent characterizations are known, one of which states that no circle intersects the edge defined by the centers of the other two circles.

The authors generalize one of these characterizations to three dimensions via a conjecture, aiming to show that in FM-tetrahedra the analogous geometric non-intersection property holds between a sphere and the opposite face. Establishing this would align the 3D case with the known 2D behavior and would simplify density computations by making solid-angle decompositions exact.