I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :

For example :

1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs