On a theorem of Helly (Q1922315)

Let \(K\) be a compact convex set in Euclidean \(d\)-space. The symbol \({\mathcal H}_K\) denotes the set of all mappings \(f:K\to K\), and \({\mathcal F}_K\) denotes the subfamily of continuous mappings. For every \(f\in {\mathcal H}_K\), the convex hull of the set of all points of the form \({d\over {d+1}}x+ {1\over {d+1}} f(x)\), where \(x\in K\), is denoted by \(K(f)\). The author proves that \(\bigcap\{K(f)\); \(f\in{\mathcal F}_K\}\neq \emptyset\). Moreover, he shows that if \(K\) is a convex polytope, then \(\bigcap\{K(f)\); \(f\in{\mathcal H}_K\}\neq \emptyset\). The author also presents a proof of Helly's theorem based on the theorem of Brouwer.