Yes. If $K \subset \Omega$ is compact, let $\text{dist}(K, \Omega^c) > r > 0$. Then for any $p \in K$ and convex function $f$, $f(p)$ is bounded by the average of $f$ over the ball of radius $r$ centred at $p$, and thus by a constant (depending only on $r$) times $\|f\|_{L^2(\Omega)}$.
Robert Israel
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