I found a function that associates each nonempty open bounded subset of $\Bbb R^n$ to a positive real number number larger than $0$ and less or equal to $1$ that is invariant under translation rotation or scaling, i.e. open bounded subsets with same shape will have the same value.
My question is if this could be useful and if so, where. For example, the value of the shape of the circle is $\pi/4$, and the value of a convex interval is $3/4$.
EDIT:
The function is the following, in a $n$ dimensional euclidean space, and $U$ an open bounded subset:
$$Q(U)=\frac{\inf\{\int_Ud(x,y)\mathbb dx\mid y\in\Bbb R^n\}}{\iint_U d(x,y)\mathbb dx\mathbb dy}\int_U\mathbb dx$$
where $d$ is the euclidean metric.
This depends to a large degree on whatever other general properties this association has. What kinds of regions get a small value and what kinds get a large value?
If it appears to be somewhat arbitrary, then it's probably not of much use. But if it, say, turned out to reliably measure how complicated the boundary of the region was (and it corresponded well with human intuition about this concept), then it could defnitely be useful.