Volume of the intersection of two lp balls.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 &lt; p &lt; q ≤ 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 &lt; 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 &gt; 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 .

Question

Volume of the intersection of two lp balls.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  &amp;lt;  p  &amp;lt;  q  ≤  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  &amp;lt;  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 &amp;gt; 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  .

Jimena Torres · Accepted Answer

Explanation:In order to realize that this is a difficult problem consider the case   d  =  2,   p  =            1      2      ,   q  =            3      2       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.

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.

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