If $S_1 = \{ x \in \mathbb{R}^2 : \|x\| = 1 \}$ is it possible to construct a function $f: S_1 \to S_1$ that is continuous and for all $x, y \in S_1$ we have $|f(x)-x| \ge \dfrac{1}{10}$ and $|f(y) - f(x)| \ge \dfrac{11}{10} |x-y|$ ?
The first I think that garantees no fixed point, but in my mind I cannot create a function that is a expansion because it has to be in $S_1$.