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Assume that a function $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is of class $C^2$, convex if necessary, $x \in \mathbb{R^n}$, $r>0$ and $z \in B(x, \frac{r}{2})$ (where $B$-open ball in $\mathbb{R^n}$).

Does there exist a constant $C>0$ depending only on $n$ such that for $$ S_z=\{y: \frac{r}{4} \leq |y-x|\leq \frac{r}{2}, Df(z) \cdot (y-z) \geq \frac{1}{2} |Df(z)| |y-z| \}$$ the following holds: $$m(S_z) \geq C r^n,$$ where $m$- is the Lebesgue measure on $\mathbb{R^n}$?

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