Theorems of Borsuk-Ulam Type

Abstract: The Borsuk-Ulam Theorem states that for any continuous function f from S^n to R^n there is some x in S^n such that f(x) = f(-x).   Replace S^n by the boundary of some open set A of E=R^{n+1} and replace R^n by some n dimensional manifold B. The conclusion of the theorem remains, with the pair x, -x replaced by some x,y on the boundary whose convex combinations contain some fixed point z in the interior of that open set. Indeed there is a topological structure to all such solutions when the z is considered a variable. If B is not a manifold, the conclusion fails. However if we allow for a finite subset x_1,…., x_n such that z is in the convex hull of the x_i, then the conclusion holds again. This is related to principal-agent situations studied in economics.

Date: Jun 27, 2019 at 16:00:00 h
Venue: Beauchef 851, Torre Norte, Piso 7, Sala de Seminarios John Von Neumann.
Speaker: Robert Simon
Affiliation: London School of Economics, Reino Unido
Coordinator: Prof. Juan Peypouquet
Abstract:
PDF

Posted on Jun 19, 2019 in Dynamical Systems, Optimization and Equilibrium, Seminars