Comparing interior angles and dihedral angles in tetrahedra

Question

Let $S\subset\Bbb R^3$ be a tetrahedron (not necessarily regular, just the convex hull of any four points in general position).

Let $v,e,\sigma\subset S$ be a vertex, an edge and a face of $S$, so that $v\in\sigma, v\in e$, but $e\not\in \sigma$ (see the image). Is it true that the interior angle $\alpha$ of $\sigma$ at $v$ is at least as large as the dihedral angle $\beta$ of $S$ at $e$ (the angle between the incident faces)?

If true, I am intrerested in a short and neat proof.

Answer 1

It turned out to be not true.

Consider the tetrahedron with vertices $$v=(0,0,0),\;w_1=(1,0,0),\;w_2=(0,1,0)\quad\text{and}\quad w_3=(0,0,1).$$ The dihedral angle at the edge $\mathrm{conv}\{v,w_i\}$ is $90^\circ$. But the face $\mathrm\{w_1,w_2,w_3\}$ is a regular triangle, having all interior angles $60^\circ$.

Answer 2

I don't know what you can assume about $\beta$, but my view is that the definition of dihedral angle is the smallest angle between two rays starting from a point of $e$, where each ray lies in one of the planes adjacent to the common edge $e$. Since $\alpha$ is one such angle, then $\alpha \ge \beta$.