## Expanding distances on a sphere

This came up in a discussion at AIM.

Let $\mathbb S^{n-1}\subset \mathbb R^n$ be the unit sphere. Suppose that $f\colon \mathbb S^{n-1}\to\mathbb R^n$ is a continuous map which does not decrease distances: that is, $|f(x)-f(y)|\ge |x-y|$ for all $x,y\in \mathbb S^{n-1}$. By the generalized Jordan theorem the complement of $f(\mathbb S^{n-1})$ has two components, precisely one of which, denoted $\Omega$, is bounded.

Prove that $\Omega$ contains an open ball of radius 1.