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Hypercubes circumscribed in hyperspheres: a constant growth ratio for volumes at large dimensions

N. Shabot-Marcos·A. Sandoval-Villalbazo
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Abstract

This note shows that an interesting property arises when considering the relation between the hypersphere volumes at dimensions $n+1$ and $n$, if the hyperspheres circumscribe unitary hypercubes in $n+1$ and $n$ dimensions, respectively . In the limit of large $n$, the ratio $\frac{V_{n+1}}{V_{n}}$ is constant and equal to the irrational number $\sqrt{\frac{πe}{2}}$. Some consequences of this result are explored.

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N. Shabot-Marcos, A. Sandoval-Villalbazo. 2004-11-04. Hypercubes circumscribed in hyperspheres: a constant growth ratio for volumes at large dimensions. https://arxiv.org/abs/math/0411106

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