In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.

In two dimensions, two such shape measures are the minimum angle of a triangle and its aspect ratio, i.e. the quotient of the radii of insphere and circumsphere.

While many of these shape measures naturally generalize to higher dimensions, and are documented in literature for arbitrary dimension, I haven't found any source which relates the minimum solid angle of a simplex with any such shape measure in arbitrary dimensions. It is "obvious" that simplices with small solid angles at the corner vertices are degenerate, but I haven't found any source on this literature.

**Question or reference request:** Can you relate the minimum solid angle of a $d$-dimensional simplex with its aspect ratio for arbitrary $d$?

A possible answer would generalize Theorem 6.1 of "A. Liu and B. Joe. Relationship between tetrahedron shape measures, BIT, 34 (1994)" which states:

For any tetrahedron $T$ we have $\sqrt{3}/24 \rho^2 \leq \sigma_{\min} \leq (2/(3^{1/4})) \sqrt{\rho}$, where $\sigma_{\min}$ is the minimum solid angle of $T$ and $\rho$ denotes the aspect ratio of $T$.