TITLE: Are homeomorphisms convex-preserving?
QUESTION [8 upvotes]: If $A\subset \mathbb R^n$ is a convex subset and $A$ is homeomorphic to a subset $B\subset R^n$, can I also say $B$ is convex? Any counterexamples?

REPLY [10 votes]: My preferred is $\varphi:(x,y)\mapsto (x,y^3)$. It is a homeomorphism ($\varphi^{-1}:(x,y)\mapsto (x,\sqrt[3]{y}$) and the image of the first diagonal is $y=x^3$ which is not convex.