I'm interested in the following notion of density for a (convex) set $X$ at a point $x$, defined as
$$D(x) = \lim_{r \to 0^+}\frac{\mu(X \cap B(x,r))}{\mu(B(x,r))}.$$
Here, $B(x, r)$ is a ball centered at $x$ with radius $r$, and $\mu$ is the Lebesgue measure. In particular for a polygon, this can be used to define the interior angle at a vertex.
I'm quite sure I've seen this before, but I can't recall where. Googling I was only able to find this question which at least confirms it's a known thing. Does anyone know of any resource that talks about this value and perhaps proves some basic properties?