Finding solutions with the Borsuk-Ulam theorem

Abstract

Many existence theorems in combinatorics and geometry reduce to the Borsuk-Ulam theorem or its generalizations. We study how these existence arguments can be turned into effective numerical methods when a simple example with a single known solution is available. The idea is to deform this example into the given problem and follow its solution during the deformation. We establish conditions under which this procedure reaches a solution and estimate the work needed for a prescribed accuracy. Mass partition problems illustrate the method for both continuous objects and finite point sets. In a fixed dimension, each step for point sets requires work proportional to the number of points; we give conditions under which the entire computation has this linear cost. We also study how changing the initial example can lead to other solutions and support a search for all of them.