Volume of the intersection of the unit ball with a polyhedral cone

Question

Given vectors $x_1,...,x_n\in\Bbb R^d$. The conic span of these vectors is

$$\mathrm{cone}\{x_1,...,x_n\}:=\{\alpha_1 x_1+\cdots +\alpha_n x_n\mid \alpha_1,...,\alpha_n\ge 0\}.$$

Question: Is there a "simple" explicit formula for computing the volume of $\mathrm{cone}\{x_1,...,x_n\}\cap \Bbb B^n$, where $\Bbb B^n$ is the unit ball centered at the origin?

$\mathrm{Vol}$ indicates the volume that I am interested in.

Answer 1

A theorem by Ribando (Measuring Solid Angles Beyond Dimension Three, Discrete Comput Geom 36:479–487 (2006)), which is a rediscovery of a result of Aomoto (Analytic structure of Schlafli function, Nagoya Math. J. 68:1-16 (1977)), is described by Beck, Robins and Sam (Positivity Theorems for Solid-Angle Polynomials, Beitrage zur Algebra und Geometrie, Vol. 51, No. 2, 493-507 (2010)) as a way to compute the solid angle at the vertex of a polyhedral cone. Ribando's result is stated as follows:

Let $\Omega \subseteq \Bbb{R}^n$ be a solid-angle spanned by unit vectors $\lbrace v_1 , \dots , v_n \rbrace$, let $V$ be the matrix whose ith column is $v_i$ , and let $\alpha _{ij} = v_i \cdot v_j$ as above. Let $T_{\alpha}$ be the following infinite multivariable Taylor series: $$T_{\alpha} = \dfrac{det \ V}{(4 \pi )^{n/2}} \sum _{a \in \Bbb{N}^{{n \choose 2}}} \left[ \dfrac{(-2)^{\sum _{i < j} a_{ij}}}{ \Pi _{i<j} a_{ij}!} \Pi _{i} \Gamma \left( \dfrac{1 + \sum _{m \neq i} a_{im}}{2} \right) \right] \alpha^{a}$$ The series $T_{\alpha}$ agrees with the normalized measure of solid-angle $\Omega$ whenever $T_{\alpha}$ converges.

Here the solid angle refers to the limit as $r \rightarrow 0$ of the volume of the intersection of the polyhedral cone with with the ball of radius $r$ divided by the volume of the ball of radius $r$ .