A Proof of Bárány's Theorem

Žana Kovijanić

We give a new proof of the following theorem of I. Bárány and L. Lowasz: Let $\Cal S_1,\Cal S_2,\dots,\Cal S_{d+1}$ be finite nonempty families of convex sets from $R^d$ and suppose that for any choice $C_1\in\Cal S_1,\dots,C_{d+1}\in\Cal S_{d+1}$ the intersection $C_i$ is not empty. Then for some $i=1,\dots,d+1$ all the sets in family $S_i$ have a common point.