Let us develop the question:

Let us focus on finite real vector space, equiped with a norm. A priori, one does not make the hypothesis that the norm is derived from a scalar product.

Which structural condition could be asked on such space that is equivalent to say this space is euclidean?

Example: in such normed space, given two vectors u and v of same length, there is a global isometry T, such that u=T(v). By global isometry T, it is meant a bijection that conserves the norm/length.

Regarding 3 (the direction I am currenty exploring):

It would be interesting to get a complete demonstration for that or a counter example.

The Mazur–Ulam theorem, applied to such T that maps zero vector to itself, implies that T is linear. This is a start.

Even such set of T transformations is a group (easy to check). How to move forward?

Some hint: since the pair of equal length (u,v) can be as closed as possible ( norm(u-v) < epsilon ), such group of T transformations should be continous. But how to prove it?

Then, how to use such constraint to move forward?Another hint: target the conclusive step that the polarization identity is necessarly implied.

Normed linear spaceschapter VII, §2, p. 151f in the third edition for many characterizations of Hilbert spaces. – Theo Buehler Sep 4 '12 at 4:32