On face angles of tetrahedra with a given base

Abstract

Let us consider the set $\Omega (△ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in ${\mathbb{E}}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $\Sigma (△ABC)$ the set $\left\{(\cos \overline{\alpha },\cos \overline{\beta },\cos \overline{\gamma })\in {\mathbb{R}}^3|ABCD\in \Omega (△ABC)\right\}$, where $\overline{\alpha }=\angle BDC$, $\overline{\beta }=\angle ADC$, and $\overline{\gamma }=\angle ADB$. The paper is devoted to the problem of determining of the closure of $\Sigma (△ABC)$ in ${\mathbb{R}}^3$ and its boundary.

42 pages, 39 figures. Typos corrected