**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?$$|f(a)-f(b)| \le |f(a)-f(c)| \le |f(b)-f(c)|$$