On Borsuk–Ulam theorems and convex sets

The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem Adams, Bush and Frick [1] that if f : S 1 → R 2 k + $f: S^1\rightarrow {\mathbb {R}}^{2k+1}$ continuous function on the unit circle S1 in C ${\mathbb {C}}$ such ( − z ) = $f(-z)=-f(z)$ for all ∈ $z\in S^1$ , then there finite subset X diameter at most π / $\pi -\pi /(2k+1)$ (in standard metric which has circumference length 2π) convex hull $f(X)$ contains 0 $0\in .