Topological, geometric, and combinatorial aspects of metric thickenings
The geometric realization of a simplicial complex equipped with the 1-Wasserstein metric of optimal transport is called a simplicial metric thickening. We describe relationships between these metric thickenings and topics in applied topology, convex geometry, and combinatorial topology. We give a geometric proof of the homotopy types of certain metric thickenings of the circle by constructing deformation retractions to the boundaries of orbitopes. We use combinatorial arguments to establish a sharp lower bound on the diameter of Carathéodory subsets of the centrally-symmetric version of the ...
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