**Motivation**: I need to find a mapping from $n$-dimensional Euclidean space to real numbers such that the distance between each pair of points in the quoted space is relatively-preserved after the application of the mapping.



> 
> **Question**: Given $a, b, c \in \mathbb{R}^{n}$ and assuming that $||a-b|| \le ||a-c|| \le ||b-c||,$
> what can the mapping $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be
> such that the following property holds?
> 
> 
> $$|f(a)-f(b)| \le |f(a)-f(c)| \le |f(b)-f(c)|$$
> 
> 
> 



Is there a mapping from Euclidean space to real numbers which relatively preserves distance?
If I understood correctly, such a mapping must be constant if $n\geq 2$.


Permuting the names of the variables, the condition implies that $f$ must send every equilateral triangle in $\mathbb{R}^n$ to an "equilateral triangle" in $\mathbb{R}$, which can only be a single point. Since every pair of points in $\mathbb{R}^n$ forms one side of an equilateral triangle, the mapping must collapse every pair of points, and hence be constant.


