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

Summer
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  • See http://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbb. Note that "convex" implies "star-shaped." – Cheerful Parsnip Nov 19 '11 at 01:10

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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.

Anonymous
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