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Let $B \subset \mathbb{R}^n$ be a collection of $n$ (not necessarily independent) unit vectors which we will label $v_1,\ldots, v_n$ for convenience. The cone $K_B \subset \mathbb{R}^n$ associated to $B$ is the non-negative linear span of $B$, i.e., $$K_B = \lbrace r_1v_1 + r_2v_2 + \ldots + r_nv_n~|~r_j \geq 0 \rbrace. $$ Let $\mathbb{S}^n$ denote the unit $(n-1)$-sphere defined as usual by $$\mathbb{S}^n = \lbrace v \in \mathbb{R}^n ~|~ \|v\| = 1\rbrace.$$

Question:

Is there a nice formula known for the ratio $$\angle B = \frac{\text{Vol}(K_B\cap\mathbb{S}^n)}{\text{Vol}(\mathbb{S}^n)}?$$

Where $\text{Vol}$ refers to $(n-1)$ dimensional volume and nice means "directly involving the coordinates of the vectors in $B$"? The motivation comes from the trivial case $n=2$: when $B = \lbrace v_1, v_2\rbrace \subset \mathbb{R}^2$ then the fraction of the unit circle's perimeter lying within the cone spanned by $v_1$ and $v_2$ can immediately be recovered from the inner product (which of course directly involves coordinates): $$ \angle B = \frac{1}{2\pi} \cos^{-1}(v_1\cdot v_2).$$

I assume this is an extremely well-studied problem, but all my google searches so far have only yielded high school trigonometry so I am obviously missing some keywords. All help is appreciated!

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3  
See mathoverflow.net/questions/7470/… –  Ryan Budney Jul 21 '12 at 3:48

3 Answers 3

In principle, one can compute the volume of a spherical polyhedron by first dividing it into simplices (e.g. by barycentric subdivision), and then computing the sum of the volumes of the simplices.

Inductive formulae for volumes of spherical simplices go back to Schlafli; see also Peschl.

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As I wrote in KvanTTT's question in math.stackexchange, see Ribando's paper: http://students.mimuw.edu.pl/~tt249057/other/ksiazki%20i%20papery/ribando-solidangles.pdf

There seems to be no closed formula, but rather a Taylor series expression.

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There are formulas for 2D and 3D dimensional cases in my question on math.stackexchange through the arctan and I still can not generalize it on 4D and higher dimensions.

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