Let $D$ be a smooth domain of $\mathbb{R}^d$. Let $\partial D$ denote the boundary of $D$. We denote by $B(x,r)=\{y \in \mathbb{R}^d \mid |y-x|<r\}$ the Euclidean ball centered at $x$ with radius $r$

For $x \in \partial D$, we consider the following limit: \begin{align*} \lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz) \end{align*} Here, $m$ denotes the Lebesgue measure on $\mathbb{R}^d$.

Does this limit exists?

In my impression, this seems to converge to the inward (unit) normal vector at $x$.