I'm taking a data structures course, and recently learned about the k-d tree, which partitions the space by dividing it with a hyperplane.
I'm wondering if there exists a data structure that partitions the space into "inner" and "outer" regions with hyperspheres that connect the nodes to their nearest node. This allows for a more intuitive way to find nodes closest to a certain point contained within a spherical region.
Each node would need to track its position in the space, a radius/distance to the nearest node upon insertion, and its inner- and outer- child nodes.
This naturally lends itself to a binary tree-like structure in which each node has an inner and outer child, and red-black trees (or other mechanisms) can keep this balanced. It has an advantage over k-d trees in that each link does not have to divide through an orthogonal hyperplane and can instead just use a hypersphere in an identical manner to the other nodes.
Similar data structures I've found to this idea include M-trees, but those have some major differences from what I had imagined, because by using a red-black tree as the underlying structure, there is no need to set a limit on the number of circles in a region. There are also ball trees and vantage-point trees, but those vary as well.
Here is a (very crude) Google Slides presentation detailing some of what I had in mind.
What is this data structure called?