# Is there a mapping from Euclidean space to real numbers which relatively preserves distance? [closed]

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?

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

• You have asked in the wrong forum. Jun 22, 2021 at 14:48
• This is definitely not a question for MathOverflow. Jun 22, 2021 at 20:26

## 1 Answer

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.