Measure of Sphere Surface inside Concentric Ellipsoid (exact or lower bound)

Consider the space $\mathbb{R}^n$, a sphere $\sum_{k=1}^n x_k^2=1$ and an ellipsoid $\sum_{k=1}^n a_k x_k^2 \leq 1$. Note that both sets are origin centered, and the ellipsoid is axis-aligned. Suppose the two sets do have partial overlap. I am interested in a closed form expression that indicates what fraction of the sphere's surface belongs to the intersection. Formally, consider the following definitions: \begin{eqnarray} \mathcal{S}&=&\{\boldsymbol{x} \in \mathbb{R}^n \,|\, \sum_{k=1}^n x_k^2=1\} \\ f(\boldsymbol{x},\boldsymbol{a}) &=& \begin{cases} 1 & \sum_{k=1}^n a_k x_k^2 \leq 1 \\ 0 & \mbox{otherwise} \end{cases} \\ p_n(\boldsymbol{a}) &=& \frac{\int_{\mathcal{S}} f(\boldsymbol{x},\boldsymbol{a}) \, d \boldsymbol{x}}{\int_{\mathcal{S}}\, d \boldsymbol{x}} \,, \end{eqnarray} I am interested in computing $p_n(\boldsymbol{a})$. Not sure if closed form $p_n(\boldsymbol{a})$ is possible in general, but at least for $n=2$ (sphere being the unit circle) I was able to compute it with Mathematica (using polar representation): $$p_2(\boldsymbol{a}) = \frac{1}{2\pi} \int_{-\pi}^\pi \begin{cases} 1 & a_1 \cos^2(\theta)+ a_2 \sin^2(\theta) \leq 1 \\ 0 & \mbox{otherwise} \end{cases} d \theta = \frac{1}{\pi} \arccos(\frac{2 - a_1 - a_2}{|a_2-a_1|})$$

If a closed form expression for the exact $p_n(\boldsymbol{a})$ is not possible for general $n$, it is also very helpful to have a closed form lower-bound, i.e. a function $q_n(\boldsymbol{a})$ such that $q_n(\boldsymbol{a}) \leq p_n(\boldsymbol{a})$.

• Thank you! How about a lower-bound on $p_n$? – Golabi May 3 '18 at 17:15
• I was responding to the title (which asks about volume), but as I read the body you are looking for "what fraction of the sphere's surface belongs to the intersection". Since you set coordinates in $\mathbb R^n$, this is especially ambiguous. – hardmath May 3 '18 at 17:55