Is it true that every convex set of the Euclidean space is the sublevel set of some convex function?

Question

Let $C \subset \mathbb{R}^n$ be a convex set.

Is it true, that there exists a convex function $f$ such that

$C = \{x | f(x) \leq a\}$ for some $a \in \mathbb{R}$

Answer 1

No, the claim as written is false. In dimension $n=1$, convex functions are continuous, so if $f$ is convex then $C$ would have to be closed. So as a counterexample, let $C=(-1,1)$.

(I don't know whether a slight change could fix the claim to be true.)