I'm working on sphere packings. When I write, I'm confused with basic definitions. I'm hesitating between the terms "sphere", "ball" or "oriented sphere".
For example, on the wikipedia page of circle packing theorem
A circle packing is a connected collection of circles whose interiors are disjoint.
But rigorously, a circle (1-sphere) is the boundary of a disk (2-ball). A sphere has no interior, a ball has. The interior is important for the definition of packings.
The problem becomes serious if I want to include hyperplanes as generalised spheres, or if I want to use the usual exterior as the interior (negatively curved balls)
I also read in literature definitions like
A sphere packing is a collection of balls with disjoint interiors
I prefer this one, but why not call it a "ball packing" directly as here? I tend to use "ball packing", but since almost every others is using "sphere packing", I'm wondering if I missed something important by calling these objects "balls". (For example, google with "Apollonian ball packing" returns no result.)
Another solution is "oriented sphere", as in Lie sphere geometry. Then my question is, what's the difference (in practice) between a sphere with an orientation, and a ball with an interior?