Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be an affine function, and let $S\subseteq \mathbb{R}^n$ be convex.
Let $rint(\cdot)$ be the mapping of a set to its relative interior.
Show that: $$ rint( f(S)) = f (rint(S)) $$
As I'm struggling with geometric proofs, I'd be especially interested in how the geometric intuition behind this helps formulating the proof (finding the right ansatz, translating intuition into formulae,...)
I guess the main idea is that the affine hull of the interior is the same as the affine hull of the set itself, and the affine hull is invariant under affine functions.
But no matter how I try to handle it, it seems to always degenerate into a lengthy messy proof with billions of technicalities.