5
$\begingroup$

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?

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

$\endgroup$
2
  • 2
    $\begingroup$ You have asked in the wrong forum. $\endgroup$ – Gerald Edgar Jun 22 at 14:48
  • $\begingroup$ This is definitely not a question for MathOverflow. $\endgroup$ – Hollis Williams Jun 22 at 20:26
10
$\begingroup$

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.

$\endgroup$
0

Not the answer you're looking for? Browse other questions tagged or ask your own question.