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What is the $n$-dimensional volume of the intersection of a unit hypercube (side length 1) and an $L_p$ ball with radius $r$, both centered at the origin, with $p > 0$? [This includes the $L_p$ seminorms where $0 < p < 1$].

This question is a generalization of intersection of hypercube and hypersphere. I am wondering if there is an analytical and/or recursive solution similar to the accepted answer there, which is given for the specific case when $p = 2$. The question is also similar to volume of the intersection of two lp balls, which does not have an accepted answer, though my question has different constraints and may be easier to solve analytically.

When $r \geq \frac{1}{2}||\mathbf{1}||_p$, the hypercube will be contained within the $L_p$ ball and the volume $V$ will be 1. When $r \leq \frac{1}{2}$, then the ball will be contained within the hypercube and

$V = 2^n r^n \frac{\Gamma\left(1 + \frac{1}{p}\right)^n}{\Gamma\left( 1 + \frac{n}{p} \right)}$,

where $\Gamma$ is the gamma function, as shown here but scaled by $r^n$ for non-unit balls.

What is the volume when $\frac{1}{2} < r < \frac{1}{2}||\mathbf{1}||_p$?

David Slater
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    The recursion (following the accepted answer literally) is $$V_n(r)=2\int_0^{\min{r,1/2}}V_{n-1}\big((r^p-x^p)^{1/p}\big)~dx,\qquad V_0(r)\equiv 1.$$ – metamorphy Feb 29 '20 at 06:17
  • Thanks @metamorphy! That was much more straightforward than I anticipated. I am curious if there's a way to represent it analytically with the gamma function, as that recursion seems very similar to that used in the derivation of Lp ball volume. This would likely need $n + 1$ formulas, depending on the size of $r$, but I'm struggling to factor the recursive element out of the integral. (Something like $V_n(r) = 2 V_{n - 1}(f(r)) \int ...$). – David Slater Mar 02 '20 at 23:47

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