Embedding of spheres in euclidean space

Question

Assume $K$ is a compact subset in $\mathbb{R}^n$. Assume $f:K\rightarrow\mathbb{R}^n$ is a topological embedding such that $\mathbb{S}^{n-1}$ is contained in $K$ and $\mathbb{S}^{n-1}$ is contained in $f(K)$. Must it be true that $f(\mathbb{S}^{n-1})$ does not lie in the open ball of radius $1$?

I've been told this is a consequence of embeddings of spheres, but I don't understand why. What about for nice compact K?

Answer 1

This is obviously false: let $K$ be the closed ball of radius $2$ and define $f$ to be the map $x\mapsto\frac{1}{2}x$.