Let $X$ be a vector space, $\Omega$ a convex subset thereof and $f:\Omega \to \mathbb R$ a convex function. Then $f$ need not be bounded from below - not even if it is strictly convex, as the example of $\Omega=X=\mathbb R$ and $f:x\mapsto x+e^x$ shows.
Things change, however, if additionally $X$ is a normed space and $\Omega$ is compact: then indeed I am able to prove that $f$ cannot be unbounded from below (unless $f\equiv -\infty$, which is excluded by assumption). However, the only proof I can come up with is based on a geometrical intuition (basically, separation of the epigraph of $f$ and a point $x_0$ below it by means of a hyperplane) that can be elementarily justified only if $X$ is a separable Hilbert space (and in particular if $\dim X <\infty$) and relies instead upon Hahn-Banach in the case of general $X$. Is there a simpler way to prove this assumption?