![]() Thus, the solution proposed in (3) double-counts these volumes. For $n=3$, this can be seen in this graphic, which was generated by wolframalpha. here)Īfter my calculation consistently gave wrong results, I was forced to admit that the case (3) is more difficult than I thought, because as soon as the opening angle of the hypercaps is larger than $\pi/4$, they start to intersect along the edges of the hypercube, whereas the corners are still outside the intersection volume. For intermediate values of $x$, the intersection is given as the volume of the hypersphere minus $2n$ hyperspherical caps, for which there is also a closed form solution (e.g.In this case, the volume is simply that of the hypercube, that is, $(2A)^n$. ![]() If $x n \cdot A^2$, the hypercube is fully contained in the hypersphere.My first idea was to separate three different cases: What is the fraction of volume of the hypercube $H$ that is also inside the hypersphere $S$, that is, what is the volume of $H\cap S$?Īs calculating the fraction with respect to the hypercube is trivial by just dividing by its volume in the end, it boils down to calculating the volume of the intersection. In $n$-dimensional euclidean space, a hypercube $H$ with side lengths $2A$ is centered around the origin. this one), but as far as I can see, none quite cuts it for me. ![]() There is a number of similar questions already (e.g. ![]()
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