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.Template:R The result was further generalized by Yamabe and Yujobô to higher dimensions,Template:R and by Floyd to other circumscribed parallelepipeds.Template:R

Kakutani's theorem on 2-spheres

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

References

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