Let $f: \Bbb{R}^n \rightarrow \Bbb{R}$ be a nonnegative function that vanishes on a set $\Omega$ that is compact and has positive measure. What is the minimial amount of regularity required of $f$ to guarantee that $\Omega$ contains an open set? I'm interested in classes of the form $C^k$ or $C^{k,\alpha}$ ($k$times continuous differentiability and Hölder spaces, respectively).

$\begingroup$ One first thought is that for $n=1$ you probably need $\alpha\geq 1/2$, although I haven't thought through this in detail $\endgroup$– Yemon ChoiJun 25, 2011 at 20:34

$\begingroup$ should be "to this site and take it over" $\endgroup$– Will JagyJun 25, 2011 at 20:36

$\begingroup$ @Jagy: point taken. $\endgroup$– Viktor BundleJun 25, 2011 at 20:57
1 Answer
By the Whitney extension theorem any closed set of $\mathbb{R}^n$ can be the zeroset of a nonnegative $C^\infty$ function. And, of course, there are closed sets with positive measure and empty interior.

$\begingroup$ or more directly, a smooth nonnegative function vanishing on a prescribed closed $\Omega$ can be constructed by means of a partition of unity. $\endgroup$ Jun 25, 2011 at 20:49