Let $(\vec{w})^{\varepsilon}$ denote an open ball of radius $\varepsilon$ centered at $\vec{w}$ . The choice of open balls over closed balls is arbitrary. Let $S$ denote a set that's constrained to be the union of an open set $T$ and its boundary.
Can you define the angle at a point $\vec{v}$ in a set $S$ embedded in $\mathbb{R}^n$ as a limit of the ratio of the volume of $S \cap (\vec{v})^\varepsilon$ and $(\vec{v})^\varepsilon$ as $\varepsilon$ approaches zero? Does this still work as a definition of angle in more exotic spaces than $\mathbb{R}^n$ with the norm $L^2$ ? Since this is not the usual way we build a notion of angles, is there a specific way in which the limit-of-volumes definition is incoherent or less parsimonious than the standard one?
I think the ordinary way of defining a notion of angle uses the inner product, $I(\vec{x}, \vec{y}) = \langle\vec{x}, \vec{y}\rangle$ , which gives you distance and angle in one fell swoop. My recollection is a bit hazy. At very least, the following equality (1) can be used to define the angle between two vectors as in (2):
$$ \cos(\theta) = \frac{\langle A,B \rangle}{\sqrt{\langle A, A \rangle \langle B, B \rangle}} \tag{1} $$
$$ \theta \stackrel{def}{=} \arccos\left( \frac{\langle A,B \rangle}{\sqrt{\langle A, A \rangle \langle B, B \rangle}} \right) \tag{2} $$
What I'm considering is the following:
$$ \frac{\theta}{2\pi} \stackrel{def}{=} \lim_{\varepsilon \to 0} \frac{\left| S \cap (\vec{v})^{\varepsilon} \right|}{\left| (\vec{v})^{\varepsilon} \right|} \tag{3} $$
In (3), $2\pi$ is just chosen to make the definition line up with radians in 2 dimensions. It probably makes sense to choose a different constant in other numbers of dimensions or just use $1$.
The intuition is that we're zooming in on one point and asking how much of our set $S = (T \cup \partial(T))$ is in a circle/sphere centered at $\vec{v}$ . Because $S$ cannot be just the boundary of a triangle, for instance, and has to include some points on either side of the boundary, we're always making a choice between exterior and interior angles.
This definition also generalizes to arbitrarily many dimensions. We can, for instance, ask what the "angle" is at the one of the points of a regular tetrahedron.
It also has the somewhat strange property that any point in the interior of $S$ has angle $2\pi$ and any point completely outside it has angle $0$ .
So anyway, this is probably a horrible way to define what an angle is and requires more machinery than just an inner product. Are there additional reasons why it's bad?
Your definition is quite reasonable, cf. steradian.
You are basically defining the density of a set at a point.
I don't think you can generalize this approach to $L^2$ because it is not locally compact, so the balls cannot have finite measure (you can pack an infinite number of balls of radius $\frac14$ inside a unit ball).