For every continuous map f: Sⁿ → Rⁿ, there exists a point x ∈ Sⁿ with f(x) = f(−x). Equivalently, no antipode-preserving continuous map Sⁿ → S^(n−1) exists. Borsuk 1933. Celebrated corollaries: ham-sandwich theorem; no planar map of the…
For every continuous map f: Sⁿ → Rⁿ, there exists a point x ∈ Sⁿ with f(x) = f(−x). Equivalently, no antipode-preserving continuous map Sⁿ → S^(n−1) exists. Borsuk 1933. Celebrated corollaries: ham-sandwich theorem; no planar map of the…