I am trying to characterise the Euclidean distance $d:\mathbb{R}^{3} \times \mathbb{R}^{3}\to \mathbb{R}$, by means of functional equations. This is my current approach, but I am not able to fill the gaps and I would like to see rigorous references in the moulds of this approach.
Consider two points $P,Q \in \mathbb{R}^3$ and a right triangle $T$ with $\overline{PQ}$ as hypotenuse. For simpliticty, also consier $P$ as the origin of the system. Let's define $d(P,Q):= \overline{PQ}$. By the properties of Euclidean geometry $d$ should satisfy the following functional equations
- $d(P,RQ) = d(P,Q)$ (invariant under rotation with center in $P$).
- $d(P, \lambda Q) = \lambda d(P,Q)$ (homogeneity of degree 1).
Let the coordinates of $Q := \{a,b,c\}$, by (1), we get
(1) $d(Q) = F(a^2 + b^2 + c^2)$
Moreover, a family of homogeneous equation of degree 1 is given by
(2) $d(Q) = (|a|^p + |b|^p + |c|^p)^{1/p}$
by combining (1) and (2), we get $p = 2$, and the following formula for the distance
$d(Q) = \sqrt{a^2 + b^2 + c^2}$
The question is: are (1) and (2) sufficient to characterize the Euclidean distance?If not, what are the other constraints on $d$?
