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.

Kakutani's theorem on 2-spheres

Given a continuous function , there exist orthonormal basis of such that .

The proof relies crucially on the fact that the fundamental group of is finite: , while the fundamental group of is infinite cyclic: .

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

References