Is it true that convex function on bounded subset of $R^n$ is bounded below?

Question

If $\Omega$ is a bounded convex subset of $\mathbb{R}^n$ and $f$ is a convex function on $\Omega$, can we say $f$ is lower bounded on $\Omega$? If it is not true, can any one provided a counter example?

Answer 1

I assume that $f$ maps into the real numbers. Then it is continuous on the relative interior of $\Omega$. Using a separation argument on the epigraph proves that $f$ is bounded from below by an affine function. Hence it is bounded on the bounded set.