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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)

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  • $\begingroup$ "Affine" would be more accurate than "linear": $B(x,y)$ only contains the origin when $x$ and $y$ have equal norms. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 19, 2017 at 15:40
  • $\begingroup$ Why would it be? if in $\mathbb R^2$ where $x=(1,0) $ and $y=(0,1)$ wouldn't $B(x,y)$ be the vertical axis for a general $L_p$ norm? $\endgroup$
    – jim h
    Commented Jan 19, 2017 at 15:45
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    $\begingroup$ In your case $x$ and $y$ have equal norms, and in the Euclidean norm $B(x,y)$ is the diagonal $(t,t)$, $t\in\mathbb R$. But take say $x=(1,0)$ and $y=(3,0)$, then $B(x,y)$ will be the vertical line passing through $(2,0)$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 19, 2017 at 15:52
  • $\begingroup$ Sorry I meant to type $y=(-1,0)$, somehow thinking ``one should be on the other side'' got me confused... In any case, please ignore my last comment. Just edited affined into the question. Thanks! $\endgroup$
    – jim h
    Commented Jan 19, 2017 at 15:56

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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, 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.

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