3
$\begingroup$

Let $f: \Bbb{R}^n \rightarrow \Bbb{R}$ be a non-negative 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).

$\endgroup$
3
  • $\begingroup$ 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 Choi
    Jun 25, 2011 at 20:34
  • $\begingroup$ should be "to this site and take it over" $\endgroup$
    – Will Jagy
    Jun 25, 2011 at 20:36
  • $\begingroup$ @Jagy: point taken. $\endgroup$ Jun 25, 2011 at 20:57

1 Answer 1

4
$\begingroup$

By the Whitney extension theorem any closed set of $\mathbb{R}^n$ can be the zero-set of a non-negative $C^\infty$ function. And, of course, there are closed sets with positive measure and empty interior.

$\endgroup$
1
  • $\begingroup$ or more directly, a smooth non-negative function vanishing on a prescribed closed $\Omega$ can be constructed by means of a partition of unity. $\endgroup$ Jun 25, 2011 at 20:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.