Is there a formal definition of "between two points" in $n$-dimensional Euclidean space? I'm just wondering if there is a formal definition describing a point is "between" other two points in $n$-dimensional Euclidean space. Since I'm not an expert on geometry or topology, I haven't seen any similar concepts. Here I give my definition of "between":
Under $n$-dimensional Euclidean space, let $d(AB)$ denote the distance from point A to point B. We say point C is "between" A and B, if and only if $d(AC) < d(AB)$ and $d(BC) < d(AB)$.
It's a quite simple definition. But I can't help thinking it's sensible enough? So I try to give a second definition:
Under $n$-dimensional Euclidean space, let $l_{AB}$ denote the line segment determined by A and B. Let $S_A$, $S_B$ are the $n$-dimensional hyper-planes determined by A, B, and $S_A \bot l_{AB}$, $S_B \bot l_{AB}$. Let $D(C,S_A)$ denote the distance from point C to hyper-plane $S_A$. Then we say C is "between" A and B, if and only if $D(C,S_A) < d(AB)$ and $D(C,S_B) < d(AB)$.
It seems also feasible. In case of any ambiguity I draw a picture to illustrate them.

Please note all the descriptions above are applicable in $n$-dimensional Euclidean space, and the figure only shows an example of 2D case.
Please do not get confused with other "between" concepts maybe like: "just draw a straight line segment, then any points on it can be C". Honestly, I would like to call this case more precisely as "point C is on the line segment determined by A and B" instead of "C is between A and B", although I'm not very clear math world would agree or not. I personally don't like this way defining "between", since it seems directly coming from the 2D "between" concepts. I believe there should be a more sensible way to extend it into higher dimensional space.
About why I ask this question, it just came up to my mind someday and I like everything in my world looking certain. It seems even in a 2D or 3D (real world) case, there is not a general doubtless definition of what "between two points" is, although we clearly know what "between two lines" is in 2D and what "between two planes" is in 3D.
I'm sincerely sorry for my limited ability of expression if it gives you any confusion. If you don't like the word "between" we may change it, but you should get what I meant above.
So, in one word, do you think the above definition of "between" is sensible or do you have any better ideas?
 A: On the line, "between" can be defined using distance inequalities, as in your question. (There are other equivalent definitions.)
You ask for a generalization to spaces of higher dimension that captures the everyday sense of "between". But that's not how or why mathematicians define concepts. A formal mathematical definition comes from the need to specify a precise relationship in some particular context. We try to choose a word for that relationship that suggests the precise meaning. But the (English) word is not what's being defined.
If you are trying to solve a problem in $n$ dimensional space and discover that your circles (spheres) or parallel (hyper)planes help, then when you write up your work you define "between" that way for your purposes.
A: Neither case 1 nor case 2 seem to be more useful in general and both seem reasonable, so I would say both could be used, depending on the context. Since no definition is generally accepted, the notion of "between-ness" would need to be explicitely defined anyway. The definition of between-ness as lying on the line segment has the advantage of working in any affine space, rather than euclidean space.
