Colloquium: Tuesday, February 22, 2022. Speaker: Shira Zerbib (Iowa State University). Title: “KKM-type theorems and their applications”.
Abstract: The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner’s lemma and Brouwer’s fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Komiya, Soberon) and have been widely applied as well. We will discuss a recent common generalization of all these theorems. We will also show two very different applications of KKM-type theorems: one is a proof of a conjecture of Eckhoff from 1994 on the line piercing numbers in certain families of convex sets in the plane, and the other is a theorem on fair division of multiple cakes among players with subjective preferences.