Kakutani’s theorem for real-valued maps on a compact surface

Abstract:

Let $M$ be a compact $2$-manifold (without boundary) ${C^1}$-embedded in ${{\mathbf {R}}^3}$. Then there exists positive $\sigma$ such that, given any positive $\tau \leqslant \sigma$ and any continuous map $f:M \to {\mathbf {R}}$, there exist points $p$,$q$,$r \in M$, satisfying $\left \| {q - r} \right \| = \left \| {r - p} \right \| = \left \| {p - q} \right \| = \tau$ in the euclidean norm, for which $f(p) = f(q) = f(r)$.