Are homeomorphisms convex-preserving?

Question

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?

Answer 1

Define, for $(x,y) \in \mathbb R^2$: $$ \varphi(x,y) := (x-|y|,y) $$ Then $\varphi$ is a homeomorphism $\mathbb R^2 \to \mathbb R^2$ but the image of $\{(x,y)\mid x = 0\}$ under $\varphi$ is not convex.

Answer 2

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.