Let $D \subseteq \mathbb{R}^n$ be a compact convex subset with a nonempty interior. Then, for any $p \in \mathrm{int}(D)$, there exists a homeomorphism $F\colon \overline{\mathbb{B}^n}\to D$ which sends $0$ to $p$, $\mathbb{B}^n$ to $\mathrm{int}(D)$ and $S^{n - 1}$ to $\partial D$.
Assume that the unit ball belongs to $D$. First, Lee proves that each closed ray strating at $0$ intersects $\partial D$ at exactly one point. That is, for any $x \in \mathbb{R}^n$ where is a unique $y \in \partial D$ such that $y = tx$ for $t \geq 0$.
Then he defines a map $f\colon \partial D\to S^{n - 1}$ by $x \mapsto x/|x|$ and proves that it is a homeomorphism.
Finally, he uses this map to define $F\colon \overline{\mathbb{B}^n}\to D$ which maps nonzero $x$ such that $|x| \leq 1$ to $|x|f^{-1}(x/|x|)$ and $0$ to $0$. I understand why it is well-defined, continuous and and injective, but can't see why it is surjective. Lee says:
It is surjective because each point $y \in D$ is on some ray from $0$.
I interpret it say saying that, for any $y \in D$, there is a unique $x \in \partial D$ such that $x = ty$ for some $t \geq 0$. However, I can't see how it helps. Indeed, we need to obtain $z \in D$ such that $|z| \leq 1$ and and $|z|f^{-1}(z/|z|) = y$.
Find point $y'\in\partial D$ such that $y=|y|/|y'|y'$ then $y=F(|y|/|y'|f(y'))$.