An Elementary Deduction of the Topological Radon Theorem from Borsuk–Ulam

Abstract

The Topological Radon Theorem states that, for every continuous function from the boundary of a (d+1)-dimensional simplex into ℝ n , there exists a pair of disjoint faces in the domain whose images intersect in ℝ n . The similarity between that result and the classical Borsuk–Ulam Theorem is unmistakable, but a proof that the Topological Radon Theorem follows from Borsuk–Ulam is not immediate. In this note we provide an elementary argument verifying that implication.