Is there a name for the operation that returns the set of interior differences of a set of vectors? ie, $X\to\{\vec a-\vec b\mid \vec a,\vec b\in X\}$ 
If $ X \subseteq \mathbb{R}^2 $, denote $X'$ as the set of all interior differences, i.e.
$$X' := \{ \vec{a} - \vec{b}, \vec{a}, \vec{b} \in X \}$$

This seems to have interesting properties, like if $X$ is two touching circles, $X'$ is three touching circles of double radius and same common diameter. Obviously could generalise to higher dimensions. Am unsure how to generally approach problems like these. Thanks in advance!
 A: This is a specific case of the Minkowski Sum. If $A$ and $B$ are subsets of some set with an addition operation (maybe $\Bbb{R}^n$, maybe a vector space, maybe just a group, or even as general as a magma), then we define the Minkowski sum of $A$ and $B$ by
$$A + B = \{a + b : a \in A, b \in B\}.$$
In your case, you are Minkowski summing $X$ with its reflection $-X$.
Minkowski sums crop up all over the place, and have some nice properties. For example, in a topological group, if $A$ is closed and $B$ is compact, then $A + B$ is closed (and compactness is necessary; the sum of two closed sets may not be closed). If both are compact, then so is $A + B$.
I actually did my honours thesis on a neat construction based on the Minkowski sum. I published a paper with my supervisor, who gave a talk about it at a fixed point conference in Turkey. The interesting property is, if $A, B, C$ are closed, bounded, non-empty convex sets of a normed lienar space (e.g. $\Bbb{R}^n$), then
$$\overline{A + C} \subseteq \overline{B + C} \implies A \subseteq B.$$
The same is true without the closure, and the closure is not necessary in finite dimensions (since $A$ and $B$ are automatically compact). Essentially, you can cancel Minkowski sums when dealing with convex sets! This actually allows you to then form a new normed linear space out of the convex sets of the space you started with.
Indeed, I've just started a postdoc position, and my new project (looking at the method of cyclic projections) uses Minkowski summation again to analyse these projections.
Basically, it's a widely useful operation.
