On Borsuk-Ulam theorems and convex sets

Abstract

The Intermediate Value Theorem is used to give an elementary proof of a Borsuk-Ulam theorem of Adams, Bush and Frick that, if $f: S^1\to R^{2k+1}$ is a continuous function on the unit circle $S^1$ in $C$ such that $f(-z)=-f(z)$ for all $z\in S^1$, then there is a finite subset $X$ of $S^1$ of diameter at most $\pi -\pi /(2k+1)$ (in the standard metric in which the circle has circumference of length $2\pi$) such the convex hull of $f(X)$ contains $0\in R^{2k+1}$.