On Borsuk-Ulam theorems and convex sets

TL;DR

This paper provides an elementary proof of a Borsuk-Ulam type theorem for functions on the circle, showing the existence of a finite subset with controlled diameter whose image convex hull contains zero.

Contribution

It introduces a simplified proof of a Borsuk-Ulam theorem variant using the Intermediate Value Theorem, connecting topology with convex geometry.

Findings

Existence of a finite subset with bounded diameter on the circle

Convex hull of the image contains zero

Elementary proof approach

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 $0\in R^{2k+1}$.