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You can certainly have a set diffeomorphic to $\Bbb R^n$ but not star-shaped. For example, for $n=2$, the Riemann mapping theorem implies that any simply connected open set is diffeomorphic to the plane. More concretely, you can take a ball and just deform it a little bit so it's very badly not convex (in particular, not star-convex) but still diffeomorphic to the ball. For example, a thickened letter M in two dimensions.