Let $x,y$ be two different points in a normed space. A bisector is defined as a set $B(x,y):=\{p: ||p-x||=||p-y||\}$, i.e. points of equal distance to both $x$ and $y$ in the space.


Theorem 25 in the second paper below basically says a finite-dimensional normed vector space is euclidean if and only if all bisectors are affine subspaces. 


Is there any way to generalize the theorem to vector metric spaces? Specifically, let $d(.,.): \mathbb R^n \times \mathbb R^n \to \mathbb R\_+$ be a metric on $\mathbb R^n$. Is it true that: $d$ is euclidean if and only if all bisectors are linear subspaces?


paper: <http://www.sciencedirect.com/science/article/pii/S0723086904800094>
This paper surveyed a lot of results about the geometric properties of finite-dimensional normed spaces. 


A more general question: Many of the properties in the above survey characterize a euclidean norm given that the space is a normed space. How much can we generalize those results under the assumption of a general metric space? I'm trying to look for accessible references along these lines. Any recommendations?


(originally posted in [stackexchange](https://math.stackexchange.com/questions/2103859/characterizing-euclidean-metric-with-bisectors))



Characterizing Euclidean metric with Bisectors (and others?)
These results can be pushed quite far for general metric spaces, and Herbert Busemann did so in The Geometry of Geodesics (1955), secs. 46 and 47. He worked in the context of [G-spaces](https://mathoverflow.net/questions/34394/g-spaces-and-manifolds), where the G is for geodesic, and proved



> 
> (Theorem 47.4) If each bisector $B(a,a')$ (i.e. the locus $xa=xa'$) of a
>  G-space contains with any two points $x,y$ at least one geodesic segment
>  between then, then the space is euclidean, hyperbolic or spherical.
> 
> 
> 


