Geometric proofs of the two-dimensional Borsuk-Ulam theorem (Q734086)

The paper presents two different proofs of the classical two-dimensional Borsuk-Ulam theorem; both are based in a description of the set \(f^{-1}(0)\) for a smooth equivariant map \(f:\mathbb{S}^2 \to \mathbb{R}\) having zero as a regular value. The first one uses elementary arguments to prove the result and the second one uses properties of symplectic quasi-states.