# Minimum solid angle and aspect ratio of an $n$-simplex

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 $T$.

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Of course you cannot have an equality relating $\sigma_{\min}$ to $\rho$ because you could fix $\sigma_{\min}$ (say, in a "needle" tetrahedron) while varying $\rho$. So are you asking for tighter bounds than provided by Liu-Joe? Do they say the bounds are not tight? – Joseph O'Rourke Feb 29 '12 at 1:40
I conjecture he is asking for this in arbitrary dimension... – Igor Rivin Feb 29 '12 at 3:35
Well, I have corrected an error. Of course, the quotient is insphere to circumsphere, not the other way around. - Any falsification of this, which I don't believe to be true, would pose some conceptual problems in computational geometry for higher dimensions, so this question is indeed motivated. And yes, I am asking for this in arbitrary dimension. I made it explicit in the text body. – shuhalo Feb 29 '12 at 10:40
@Martin & @Igor: OK, I see the question now. Thanks. – Joseph O'Rourke Feb 29 '12 at 11:13
See Boris Bukh's question, Angle of a regular simplex. – Joseph O'Rourke Nov 21 '15 at 17:50