I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry.
Motivation: Suppose $A$ and $B$ live in an $n$-dimensional Euclidean space. They are each assigned the task of constructing an equilateral triangle of side length 5. Subject $A$ finds first one point $p_1$. If $n>0$, he can then find another point $p_2$ at distance 5 from $p_1$. If $n>1$, he can then find $p_3$ at distance 5 from $p_1$ and $p_2$. Then $B$ proceeds similarly. He selects a point $q_1$ (probably different from $p_1$), then finds another point $q_2$ and $q_3$. We wouldn't expect him to get stuck before constructing $q_3$, since $n$ is the same for both subjects. I will say a space is all-equal if the possibility of completing a figure is independent of the starting points chosen. Are Euclidean spaces all-equal? Are there non Euclidean all-equal spaces?
Precise statement
Definition: A metric space $X$ is all-equal if every time $S$ and $T$ are isometric subspaces of $X$ (with a selected isometry $S\to T$), the isometry extends to an automorphism of $X$.
Question 0: Is every Euclidean space all-equal?
To state the second question we must first observe some limitations. Being connected our life intervals, we cannot visit more than one connected component of our space, and being imprecise our measurements, we cannot distinguish directly between a space and its completion. Finally, existing no natural unit of measurement, the correct category to state these questions is that of spaces in which the distance is defined up to scale. That is, it takes values in a 1-dimensional module $M$ over $[0,+\infty)$, with no multiplication structure (although lengths can be tensorised to obtain areas).
More definitions: A congruence space is a pair $(X,M,d)$ where $X$ is a set, $M$ is a 1-dimensional $[0,+\infty)$ module and d is a distance in $X$ taking values in $M$. The morphisms from $(X,M,d)$ to $(Y,N,e)$ will be the subsimilarities, that is, the pairs $(f,\alpha)$ where $f$ is a function from $X$ to $Y$ and $\alpha$ is a morphism from $M$ to $N$ such that for each $x$, $x'$ in $X$ we have $e(fx,fx')\leq\alpha(d(x,x'))$. Similarities are similarly defined using an $=$ sign instead of $\leq$. Subspaces are defined by restriction, and the identity of $(X,M,d)$ is the obvious $(id_X,id_M)$. It's easy to prove that every isomorphism is a similarity. A congruence space $X$ is all-equal if every time $S$ and $T$ are similar subspaces, the similarity extends to an automorphism of $X$.
Question 1: Is every connected complete all-equal space a finite dimensional real inner-product space?
Remarks: if the connectedness hypothesis is dropped, some discrete spaces would be all-equal. If we work in the usual category of metric spaces, projective and hyperbolic spaces could be all-equal, and also Euclidean spheres, both with their inner metric and with the subspace metric.
Partial results and lines of thought:
If a real Banach space is all-equal, I think that it is finite dimensional and inner-product. This can be proved (at least in the finite dimensional case) using the John ellipsoid (see http://mathoverflow.net/questions/41211/easy-proof-of-the-fact-that-isotropic-spaces-are-euclidean), even though I don't like to introduce such a cumbersome structure.
If a Riemannian manifold is all-equal, then it is homogeneous and with constant curvature (how do we deal with the fundamental group?). If a Finsler manifold is all-equal, then I would expect it to be Riemannian (again using the John ellipsoid). I have no idea how to proceed without differential structure, and I really don't have any intuition of how the characterization could be true (only hope).
EDIT: As noted by Sergei, there are counterexamples. Generally, if $(X,d)$ is an all-equal metric space, and $p>1$, then $(X,d^{\frac 1p})$ could be an all-equal space. This is similar to the example in which However, I was thinking of spaces in which the distance from $x$ to $y$ is measured using a signal that travels form $x$ to $y$ at constant speed. Hence I require that the space have the following property:
If $x$, $z$ are points and $a$, $b$, are lengths such that $a+b>d(x,z)$, then there exists $y$ such that $a>d(x,y)$ and $b>d(y,z)$.
Which is similar to saying that the inner metric induced by the distance is not greater than the distance itself (but I can't find right now the definition of the induced inner metric).
After this tweaking, it is easy to construct approximate midpoints, then midpoints (using completeness), then segments, then complete geodesics (a segment can be extended by moving the staring point to the midpoint and the midpoint to the end). Then each pair of points can be joined by a geodesic of the expected length.
Related MO questions:
http://mathoverflow.net/questions/47882/characterizations-of-euclidean-space
http://mathoverflow.net/questions/41211/easy-proof-of-the-fact-that-isotropic-spaces-are-euclidean
http://mathoverflow.net/questions/8513/characterization-of-riemannian-metrics
