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Let $ A(n, r, x) = B^n_r(x) \cap [0,1]^n $ denote the intersection between an $n$ ball $B^n_r(x)$ with arbitrary radius $r$ and arbitrary center $x \in \mathbb{R}^n$ that intersects a unit $n$ cube $ [0,1]^n $. I'm interested in calculating $Vol_n (A(n, r, x)) $.

The case when they are concentric, $Vol_n (A(n, r, x = x_{0.5} ))$ where $ x_{0.5} = (0.5, ..., 0.5) $, is solved by several people about 20 years ago:

http://world.std.com/~dcons/Maths/siam9619.html

Is there a generalization of the above formula for arbitrary $x$?

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  • $\begingroup$ It seems this might benefit from calculating the division of the hyper ball into 2^n (or fewer) parts by orthogonal slices sharing a single point, and determining the volume of each part. You might see if that problem has also been studied. Gerhard "Picking A Problem To Pieces" Paseman, 2018.02.03. $\endgroup$
    – Gerhard Paseman
    Commented Feb 4, 2018 at 2:57

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