Given an orientable genus-0 polyhedral surface defined by n triangles, and a
set of m point sites on it, we would like to identify its 1-center, i.e., the location on the
surface that minimizes the maximum distance to the sites. The distance is measured as
the length of the Euclidean shortest path along the surface. To compute the 1-center, we
compute the furthest-site Voronoi diagram of the sites on the polyhedral surface. We show
that the diagram has maximum combinatorial complexity Θ(mn2), and present an algorithm
that computes the diagram in O(mn2 logm log n) expected time. The 1-center can then be
identified in time linear in the size of the diagram
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