# Perimeter of level sets of a smooth function

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.

• 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. – David Tewodrose Mar 27 at 10:22
• 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). – David Tewodrose Mar 27 at 15:49