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Kakutani's theorem is a result in geometry named after Shizuo Kakutani. It states that every convex body in 3-dimensional space has a circumscribed cube, i.e. a cube all whose faces touch the body. The result was further generalized by Yamabe and Yujobô to higher dimension, and by Floyd to other circumscribed parallelepipeds.
References

Kakutani, S. (1942), "A proof that there exists a circumscribing cube around any bounded closed convex set in R3", Annals of Mathematics (2) 43 (4): 739–741, doi:10.2307/1968964.
Yamabe, H.; Yujobô, Z. (1950), "On the continuous function defined on a sphere", Osaka Math. J. 2 (1): 19–22.
Floyd, E. E. (1955), "Real-valued mappings of spheres", Proceedings of the American Mathematical Society 6 (6): 957–959, doi:10.2307/2033116.

Mathematics Encyclopedia

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