In this paper we solve the problem of circumscribed cubes, frequently simply named Kakutani’s theorem. This theorem tells us, that we can circumscribe a cube around every closed bounded set in ${\mathbb R}^3$. Furthermore we describe the theory of homotopies, fundamental groups and contractible spaces needed to prove, that every double loop in the group $SO(3)$ is homotopic to a constant map. We also prove a few corollaries of Kakutani’s theorem, one of them being the extension of the problem to higher dimensions. It turns out that the answer to that problem is also affirmative. That means, that we can circumscribe a $n$-cube around every closed bounded set in ${\mathbb R}^n$. At the end we briefly summarize the history of solving Knaster’s problems and prove that in general Knaster’s conjecture is false.
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