Kakutani's theorem (geometry)

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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.

[edit] References

• S. Kakutani, A proof that there exists a circumscribing cube around any bounded closed convex set in R³, Ann. of Math. (2) 43 (1942), 739–741.
• H. Yamabe, Z. Yujobô, On the continuous function defined on a sphere, Osaka Math. J. 2 (1950), 19–22.
• E. E. Floyd, Real-valued mappings of spheres, Proc. Amer. Math. Soc. 6 (1955), 957–959.