An extremely simple proof of the K-K-M-S Theorem

Summary.

 An extremely simple proof of the K-K-M-S Theorem is given involving only Brouwer’s fixed point theorem and some elementary calculus. A function is explicitly given such that a fixed point of it yields an intersection point of a balanced collection of sets together with balancing weights. Moreover, any intersection point of a balanced collection of sets together with balancing weights corresponds to a fixed point of the function. Furthermore, the proof can be used to show -balanced versions of the K-K-M-S Theorem, with -balancedness as introduced in Billera (1970). The proof makes clear that the conditions made with respect to by Billera can be even weakened.