Given (1) a $d$-dimensional space,
(2) a $l_p$ ball of radius $r_1$, and
(3) a $l_q$ ball of radius $r_2$, where $0<p<q \leq 2$,
(4) both balls are centered on the origin.
Please can someone help me in finding the volume of the intersection of these two balls?
In low dimensional space, for example $d < 10$, I can compute the volumn by applying monte carlo simulation. However, the monte carlo method does not work in high dimensional space, e.g. d > 100, because the number of samples required is huge, which is beyond the computational power. Therefore, I wish to get the formula of the volume with respect to $d$, $p$, $q$, $r_1$ and $r_2$.
Thank you very much for your help!!!
A hint:
In order to realize that this is a difficult problem consider the case $d=2$, $p={1\over2}$, $q={3\over2}$ with variable $r_1$, $r_2$. Draw a figure! Computing the area of the intersection of the two balls involves cases depending on the sizes of $r_1$, $r_2$, then solving unfriendly equations, and finally tricky integrals. Good luck!