## M. Longinetti - S. Naldi - A. Venturi

# R-hulloid of the vertices of a tetrahedron

created by naldi on 05 Jun 2024

[

BibTeX]

*preprint*

**Inserted:** 5 jun 2024

**Last Updated:** 5 jun 2024

**Year:** 2024

**Abstract:**

The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, 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 $O^*\not \in V$ 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 $\rho$ such that $V$ is a $\rho$-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.