Let $X$ be a non-compact subset of a normed vector space. Let $f : X \to \mathbb{R}$ be a differentiable convex function that is bounded from below. Given that $X$ is not compact:
1) Is there a simple sufficient condition on $f$ that guarantees that the infimum is attained within $X$, i.e., that there exists some $x \in X$ such that $f(x) = \inf_{y\in X} f(y)$?
2) Is there a simple sufficient condition on $f$ that guarantees that the infimum is attained within every non-empty convex subset of $X$?