Could the linear transformation of a convex closed set not closed?

Question

Given a convex closed set $S \in \mathbb{R}^n$, could its linear transformation $AS$, where $A \in \mathbb{R}^{m\times n}$ not closed? Is there any example for that?

Answer 1

Yes. Look at the set $$C:= \{(x,y)\in \mathbb{R}^2:x>0, \, y\ge \frac{1}{x}\}$$ and use the map $$(x,y)\mapsto x$$ (the projection onto the first factor), or, if you want a map from $\mathbb{R}^2$ into itself,

$$(x,y)\mapsto (x,0)$$

You should be able to verify that $C$ is closed and convex, but the image of $C$ under the projection is just the positive part of the real line, which is open in $\mathbb{R}$ and at least not closed as a subset of $\mathbb{R}^2$