The Computation of Antipodal Fixed Points

Abstract

In this chapter we consider the problem of finding zero points of a continuous function f from the n-dimensional symmetric cube C ⁿ into ℝⁿ.It will be shown in a constructive way that there exists a zero point of f in C ⁿ if for every ∈ bd(C ⁿ), f (−x) ≠ α f (−x) for all α > 0. It will be argued that this result is so strong that it implies several powerful fixed point theorems, including Borsuk-Ulam’s theorem and Brouwer’s fixed point theorem. We will introduce two algorithms to compute zero points of the continuous function. One is an integer labeling algorithm and the other a vector labeling algorithm. The integer labeling algorithm will lead to a constructive proof for Tucker’s theorem. The algorithm we introduce here is a modification of the algorithms developed by Todd and Wright [1980], Freund and Todd [1981], and van der Laan [1984] in the sense that the modified algorithm can start with an arbitrarily chosen point in C ⁿ, whereas the existing methods can only start with the point 0ⁿ. The modified algorithm will be built upon a new triangulation of C ⁿ, called the AS-triangulation.