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 f : S 2 → R {\displaystyle f:S^{2}\to \mathbb {R} } , there exist orthonormal basis u , v , w {\displaystyle u,v,w} of R 3 {\displaystyle \mathbb {R} ^{3}} such that f ( u ) = f ( v ) = f ( w ) {\displaystyle f(u)=f(v)=f(w)} . The proof relies crucially on the fact that the fundamental group of S O ( 3 ) {\displaystyle SO(3)} is finite: Z 2 {\displaystyle \mathbb {Z} _{2}} , while the fundamental group of S 1 {\displaystyle S^{1}} is infinite cyclic: Z {\displaystyle \mathbb {Z} } . This result easily implies the theorem on inscribing convex bodies in cubes.
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