Function and colorful extensions of the KKM theorem (Q2027811)

Let \(\Delta (A)\) be an \(n\)-dimensional simplex, where \(A = \{a_0, a_1, \dots, a_n\}\) is the set of its vertices. A set \(C \subset \Delta (A)\) is called a corner set at vertex \(a \in A\) if \(a \in C\) and \(\Delta (A) \setminus C\) is a union of convex sets each containing \(\Delta \big(A \setminus \{a\}\big).\) The main result of the paper is the following: Let \(\Delta (A)\) be an \(n\)-dimensional simplex and, for every \(i \in \{0, 1, \dots, n \}\) let \(\{F_a^i: a \in A\}\) be a KKM family on \(\Delta (A)\) consisting of closed corner sets. Then there exists a bijection \(\sigma: \{0, 1, \dots, n\} \rightarrow A\) such that \(\{F_{\sigma (i)} ^i: i = 0, 1, \dots, n\}\) is a KKM family. Clearly the result above is closely related to the Gale generalization of the KKM priniple [\textit{D. Gale}, Int. J. Game Theory 13, 61--64 (1984; Zbl 0531.90011)]. Simple examples show that the conclusion is no longer true if we renounce to the condition that the closed sets should be corner sets. The following open problem arises naturally: Which KKM families, other than those consisting of closed corner sets, satisfy the above theorem?