  We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex
maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the
unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb
B^n$ be the set of points $\xi$ such that $\lim_{z\to \xi}\|F(z)\|=\infty$.
Then we prove that $\mathcal S$ is either empty, or contains one or two points
and $F$ extends as a homeomorphism $\tilde{F}:\overline{\mathbb B^n}\setminus
\mathcal S\to \overline{D}$. Moreover, $\mathcal S=\emptyset$ if $D$ is
bounded, $\mathcal S$ has one point if $D$ has one connected component at
$\infty$ and $\mathcal S$ has two points if $D$ has two connected components at
$\infty$ and, up to composition with an affine map, $F$ is an extension of the
strip map in the plane to higher dimension.
