As indicated by the tag, this is a reference request of a notion I thought of, but have not been able to find any literature on it. So, maybe someone here can link some resource.
Suppose there are $n$ bodies with masses $m_1, m_2, ..., m_n$, respectively. I am interested in defining a notion of center of mass, which takes into account the position and mass of the bodies, but also weighs in whether given bodies are closer together or not (form a cluster).
Let me provide an example: suppose there are 4 bodies with equal mass $m$. They are located at $(-2,0)$, $(2,0)$, $(2,1)$, $(2,-1)$. Then, using the traditional notion of center of mass, we would get that the center of mass is at $(1,0)$. The notion of center of mass I am thinking of would have a center of mass $(1+\epsilon, 0)$ for some $\epsilon>0$ because three bodies are grouped on the right hand side of the plane.
There are many ways to go about doing this. I don't think this is necessarily a far-fetched notion. So, I'd imagine some people already thought about this and established some ways to go about it, plus they showed what properties their ways exhibit. I'd appreciate if someone can share resources or share their own ideas.
The idea I thought of so far is as follows: Let $m_i$ and $p_i$ be the mass and position of body $i$. Then, let $r_{i,j}$ be the distance from body $i$ to $j$. Then, the "neighboring" mass is given by $w_i=\sum_{j}\dfrac{1}{r_{i,j}}$. Then let the mass be $m_i=m_iw_i$, that is, the original mass is weighted by $w_i$. This change will shift the center of mass towards bodies forming clusters.
If my question is too vague/broad/lacks too much rigor, please let me know and I'll try to adjust it or remove it.