TITLE: When can we extend distance-preserving maps between finite sets of points to isometries?
QUESTION [5 upvotes]: Suppose I have $n$ points $x_1, \ldots ,x_n \in \mathbb{R}^d$ where the affine span of $\{x_1, \ldots ,x_n \}= \mathbb{R}^d$. If I have another set of points $y_1, \ldots, y_n$ where $\|y_i - y_j\| = \|x_i - x_j \|$ for all $i,j \in \{1, \ldots, n\}$ where $\| \cdot \|$ is the Euclidean norm then there exists an isometry $f : \mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $f(x_i)=y_i$ for all $i = 1 , \ldots, n$. The same cannot be said for general normed spaces, for instances if we take $$x_1 =(0,0), ~ x_2 = (1,0), ~ x_3=(0,1)$$ in $(\mathbb{R}^2, \| \cdot \|_1)$ and $$y_1 = (0,0), ~ y_2 = (0.5,0.5), ~ y_3 = (0.5,-0.5).$$ Does anyone know of where to look into such results? My intuition say that for $\mathbb{R}^d$ with a smooth $\ell_p$ norm (i.e. $1<p<\infty$) we would need $2d$ points but I have no idea where to begin to look for papers in this area. Any ideas and/or references would be much appreciated.

REPLY [3 votes]: This is an amazing property of Euclidean geometry.  Given any subset $A \subseteq \mathbb R^n$ and an isometry of $A$ into $\mathbb R^n$, there is an extension to an isometry of the whole space.  And if the affine span of $A$ is the whole space, then the extension is unique.  Hyperbolic space also has this property.  In dimension 2, this is classical (as in classical Greek times) and can be thought of as underpinning the theory of triangles.
Long ago I learned all about this, and much more, in the old book
Busemann & Kelley Projective Geometry and Projective Metrics
which is likely in your university library.  They study many types of geometry; this extension property for isometry is sometimes seen, but it is rare.