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

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.

• What do you mean by simple? Simple to write down is sufficient enough, or must it be simple to compute? Commented Jun 22, 2021 at 1:20
• @ABBalbuena I need it for theoretical purpose, so I am not so much interested in computational efficiency. I am looking for a formula that is "easy to work with", but of course this is still vague. Commented Jun 22, 2021 at 1:21

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: $$T_{\alpha} = \dfrac{det \ V}{(4 \pi )^{n/2}} \sum _{a \in \Bbb{N}^{{n \choose 2}}} \left[ \dfrac{(-2)^{\sum _{i < j} a_{ij}}}{ \Pi _{i The series $$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$$ .