When does the image of a point lie in the relative interior of a convex set?

Suppose we have a continuous function $F : \mathbb{R}^n \to \mathbb{R}^n$ that is injective and such that the closure of the image of $F$, $\overline{F(\mathbb{R}^n)} \subset \mathbb{R}^n,$ is a bounded convex set.

Is it true that $F(0)$ belongs to the interior of $\overline{F(\mathbb{R}^n)}$?

Thanks for any help in advance.