The diffeomorphism of convex set

Question

For a convex open set $H$ in $\mathbb R^n$, is it diffeomorphic to $B = \{ x \in \mathbb R^n|\left\| x \right\| < 1\} $?

Answer 1

I think you would need some more conditions on your convex set in order for it to be diffeomorphic to the unit ball in general. For example, $H$ could be a line in $\mathbb R^n$ with $n>1$. Then $H$ doesn't even have the same dimension as the unit ball $B$ in $\mathbb R^n$, and so it can't be diffeomorphic to it. The closed unit ball $B=\{x∈R^n|∥x∥\leq 1\}$ (a manifold with boundary) is also a convex subset of $\mathbb R^n$, but it is not diffeomorphic to the open ball.