Suppose $M^n$ is a Riemannian manifold and $f\colon M\to \mathbb{R}$ a function on it. Take a regular point of $f$, $p$. I want to prove that there exists a positive $\epsilon$ depending on $p$ such that we have

$$\frac{\mathcal{H}^{n-1}(f^{-1}(t)\cap B_{\delta}(p))}{\omega_{n-1}\cdot\delta^{n-1}}\leq2$$

for every $t$ in $\mathbb{R}$ and $\delta<\epsilon$. Do you have any idea?



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