AN EXTREMELY SIMPLE PROOF OF THE K-K-M-S THEOREM

An extremely simple proof of the K-K-M-S Theorem is given involving on ly Brouwer's fixed point theorem and some elementary calculus. A funct ion is explicitly given such that a fixed point of it yields an inters ection 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 t he function. Furthermore, the proof can be used to show pi-balanced ve rsions of the K-K-M-S Theorem, with pi-balancedness as introduced in B illera (1970). The proof makes clear that the conditions made with res pect to pi by Billera can be even weakened.