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I'm trying to get up to speed on the Weber problem. I'm very much not an expert, unfortunately. I would really appreciate recent literature recommendations / surveys / potted histories, however incomplete. And more specifically:

  • Is there a solution for rectilinear space?
  • Are there multi-dimensional, rather than planar, solutions?
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The Fermat-Weber problem can be formulated as a conic quadratic problem (aka. SOCP). These problems can be solved in polynomial complexity using an interior-point method. See the Mosek modelling cookbook for information about conic optimization.

You can use Mosek to solve those problems.

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