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Joseph O'Rourke
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If you will excuse me substituting a polyhedron for the Riemannian manifold, this figure shows how the cut locus (red) from source $x$ is a tree. (The green arcs are equidistant from $x$.)
     2x1x1Box
     (Figure from Discrete and Computational Geometry)

Joseph O'Rourke
  • 150.9k
  • 36
  • 358
  • 958