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I've a simple question concerning the perimeter of level sets of a smooth function.

Let $f:\Omega \to \mathbb{R}$ be a smooth function defined on a bounded domain of $\mathbb{R}^n$. We set $A_s:=\{f>s\}$ for any $s \in \mathbb{R}$. By the Morse-Sard lemma, we know that for a.e. $s \in \mathbb{R}$, $\partial A_s$ is a smooth hypersurface. We denote by $I$ the set of such values $s$ and by $\mathcal{P}(A_s)$ the perimeter of $A_s$ for any $s \in I$. Does there exist C>0 such that: $$ |\mathcal{P}(A_s) - \mathcal{P}(A_{s'})|\le C|s - s'| \qquad \forall s \in I? $$

I've looked into Federer's book on Geometric Measure Theory, without any success, and I'm looking for other references in which I could find whether this question has been answered yet.

Thanks.

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  • $\begingroup$ I've also tried the book "Geometric Measure Theory and Real Analysis" edited by L. Ambrosio, but there's nothing there related to my question. $\endgroup$ – David Tewodrose Mar 27 at 10:22
  • $\begingroup$ I think now that it depends on the regularity of the boundary of the domain. If the domain has fractal boundary, we might be able to construct a smooth function $f : \Omega \to \mathbb{R}$ converging to +$\infty$ when one approaches the boundary of $\Omega$ in such a way that the level lines of $f$ approaches the boundary of $\Omega$ (so their perimeter explodes). $\endgroup$ – David Tewodrose Mar 27 at 15:49

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