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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!!!

user64411
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  • No chance for an explicit formula except for trivial cases, but if you really need something more realistic, just restate the question accordingly and someone will answer. – fedja Dec 30 '14 at 22:49
  • Thank you for your answer. – user64411 Jan 07 '15 at 04:13

1 Answers1

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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!

Christian Blatter
  • 232,144
  • Thank you for your answer. In low dimensional space, I can get the answer by 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 $r_2$. – user64411 Jan 07 '15 at 04:11