Suppose, $X$ is a Banach space with norm $||...||$, $f: [0; 1] \to X$ is a continuous function, such that $f(0) = f(1)$ and $\forall x \in (0; 1)$ and $y \in [0; 1]$ $f(x) \neq f(y)$. Do there always exist such $a, b, c \in [0; 1]$, that $||f(a) - f(b)|| = ||f(b) - f(c)|| = ||f(c) - f(a)|| > 0$?
I know that this statement is true for the Banach spaces $(\mathbb{R}, |...|)$ (by vacuity) and $(\mathbb{R}^2, ||...||_2)$ (Meyerson, Marc D. “Equilateral triangles and continuous curves”), but do not know, whether this statement is true in general.