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 of whose faces touch the body.^[1] The result was further generalized by Yamabe and Yujobô to higher dimensions,^[2] and by Floyd to other circumscribed parallelepipeds.^[3]

Kakutani's theorem on 2-spheres

Given a continuous function $f:S^{2}\to \mathbb {R}$ , there exist orthonormal basis $u,v,w$ of $\mathbb {R} ^{3}$ such that $f(u)=f(v)=f(w)$ .

The proof relies crucially on the fact that the fundamental group of $SO(3)$ is finite: $\mathbb {Z} _{2}$ , while the fundamental group of $S^{1}$ is infinite cyclic: $\mathbb {Z}$ .

This result easily implies the theorem on inscribing convex bodies in cubes.

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