continuous map on compact ellipse

Question

will there be any fixed point a continuous $f$ from the ellipse $2x^2+3y^2\le 1$ to itself? Well I think yes but in a solution of a problem hint is given that NO. Just asking to assure myself if I am not missing anything of the condition of fixed point theorem.

Answer 1

The ellipse $E=\{(x,y):x^2+3\,y^2\le1\}$ is a convex copmpact subset of $\mathbb{R}^2$. Brouwer's fixed point theorem implies that any conyinuous $f\colon E\to E$ has a fixed point.