Let $D_1$ and $D_2$ be two bounded domains in $\mathbb{C}^n$. Let $f:D_1 \to D_2$ be a proper holomorphic map and $V = \{z \in D_1: Det[f'] = 0\}$. Let $U_{\epsilon} = \{z \in D_2: dist(z, f(V)) < \epsilon \}$. Show that is $m(U_\epsilon) = O(\epsilon^2)$ as $\epsilon \to 0$ where $m$ is the Lebesgue measure?
Work so far:
Since $f$ is a proper holomorphic map and $V$ is a variety, $f(V)$ is also a variety. Thus, $f(V)$ is an analytic set, its codimension is greater than or equal to 1, and its Lebesgue measure is equal to 0. The fact that its Lebesgue measure equals 0 isn't sufficient, but maybe the codimension requirement is?
I believe my question has now been reduced to if $f(V)$ is a holomorphic variety in a bounded domain and if $U_\epsilon = \{z \in D_2: dist(z, f(V)) < \epsilon\}$, then is $m(U_\epsilon) = O(\epsilon^2)$?