 02193 J. J. P. Veerman
 Mediatrices and Connectivity
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Apr 22, 02

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Abstract. If $(X,d)$ is a connected metric space and $a$ and $b$ are points
in $X$, then the locus $L$ of the points $x$ where
$d(x,a)d(x,b)=0$ is called a mediatrix. For instance if $X$ is a
geodesic space then the geodesics emanating from $a$ and $b$
(starting at the same time, and travelling with unit speed) are
said to focus at $L$.
In an earlier paper (\cite{VPRS}), Brillouin Spaces were defined.
These are spaces in which mediatrices have desirable properties.
Most importantly: they are minimally separating. This means that
for every proper subset $L'$ of $L$ in $X$, $XL$ is disconnected,
but $XL'$ is connected.
The purpose of this note is twofold. First, we give a very simple
characterization of Brillouin Spaces, which shows that, for
example, compact, connected Riemannian manifolds are Brillouin
Spaces. Second, we give a description in terms of homology of
these mediatrices. This leads to a complete topological
classification of mediatrices in 2dimensional, compact,
connected, Riemannian manifolds.\\
KEYWORDS: Minimally separating, homology, geodesics, Riemannian
manifolds, length spaces.
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