KKM theorem

The Knaster-Kuratowski-Mazurkiewicz (KKM) theorem is a mathematical result important in fixed point and game theory. It states that if you have a simplex (a shape like a triangle or tetrahedron) and assign to each point a certain "covering" subset in a consistent way—specifically, each vertex's subset covers the opposite face—then all these subsets together must share at least one common point. Essentially, it guarantees a point that lies within all these assigned sets, which has important implications for equilibrium and optimization in economics and other fields.