Kakutani's theorem (geometry)

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 of whose faces touch the body. The result was further generalized by Yamabe and Yujobô to higher dimensions, 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 .