Apologies if this question might be trivial or has been asked already (haven't found an equivalent post), but I am trying to figure out whether the following is true:

Given two convex sets $\mathcal{X} \subseteq \mathbb{R}^n$ and $\mathcal{Y} \subseteq \mathbb{R}^n$, is $\mathcal{Z} := \{x \odot y ~|~ x \in \mathcal{X}, y \in \mathcal{Y}\}$ convex? (Here $\odot$ represents the elementwise multiplication of both vectors.)

If I had to guess I'd say "yes", because if $\mathcal{X}$ is just a point, then $\mathcal{Z}$ is a re-scaled/flipped version of $\mathcal{Y}$ which is convex, and if $\mathcal{X}$ is a line segment (i.e. we only vary a single entry) then $\mathcal{Z}$ is the union of convex sets that can only grow/shrink in one direction, which is also convex. Beyond that I am not sure...

Thank you!