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.

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