Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit $$ \lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz) $$ where
- $B(x,r)=\{y \in \mathbb{R}^d \mid |y-x|<r\}$ is the Euclidean ball centered at $x$ with radius $r$ and
- $m$ is the Lebesgue measure on $\mathbb{R}^d$,
could exist (possibly in a almost-everywhere sense).
Two more or less elementary answers possibly are
- $D$ is a Caccioppoli set and $x$ is a point of its reduced boundary and
- when $D$ has a singular part with finite (electrostatic or Newtonian) capacity (see this Q&A for reference).
Are there any other cases? Also references are welcome.
