R-hulloid of the vertices of a tetrahedron
Marco Longinetti, Simone Naldi, Adriana Venturi
TL;DR
The paper develops a 3D generalization of the convex hull via $R$-hulloids for the vertex set of a tetrahedron, introducing the critical radius $r_L(V)$ and a four-sphere touching configuration that yields an interior point $O^*$ when $R^*>r(V)$. It provides a concrete representation of ${\mathrm{co}}_\rho(V)$ for all $\rho>0$ in terms of spheres $B_i(\rho)$ with centers on special lines, and analyzes the transition to fullness through $R^*$, where ${\mathrm{co}}_{R^*}(V)$ collapses to $V\cup\{O^*\}$ or remains $V$ depending on whether $R^*$ exceeds $r(V)$. The work connects to planar Johnson’s Theorem and to unit-sphere-system results in $\mathbb{R}^3$, showing that in 3D the hulloid boundary for $\rho>R^*$ consists of four spherical caps and that $O^*$ may lie inside the tetrahedron rather than at a classical center. It also discusses uniqueness issues and illustrates examples including well-centered vs non-well-centered tetrahedra, with implications for geometric modeling and image analysis.
Abstract
The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, 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.