Properties of plane angles of tetrahedra with a given base

Let $\Omega(\triangle ABC)$ be the set of all tetrahedra $ABCD$ in three-dimensional Euclidean space with a given non-degenerate base $ABC$ and a vertex $D$ lying outside the plane $ABC$. Consider the set
$$ \Sigma(\triangle ABC)= \bigl\{(\cos \alpha, \cos \beta, \cos \gamma)\in \mathbb{R}^3 : ABCD \in \Omega(\triangle ABC)\bigr\}, $$
where $\alpha =\angle BDC$, $\beta= \angle ADC$, and $\gamma = \angle ADB$. This talk is devoted to describing the closure of the set $\Sigma(\triangle ABC)$ in $\mathbb{R}^3$.
The main results were obtained in the following paper:
E.V. Nikitenko, Yu.G. Nikonorov, On face angles of tetrahedra with a given base, 2025, arxiv:2505.22374.