02-193 J. J. P. Veerman
Mediatrices and Connectivity (40K, latex) 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\$, \$X-L\$ is disconnected, but \$X-L'\$ 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 2-dimensional, compact, connected, Riemannian manifolds.\\ KEYWORDS: Minimally separating, homology, geodesics, Riemannian manifolds, length spaces.

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