# What fraction of a sphere's volume lies within a cone?

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!

• – Ryan Budney Jul 21 '12 at 3:48