Abstract
We study hidden geometry of Bier spheres Bier(K) = K ?? K? by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to Bier(K) a natural coarsening Fan(K) of the braid fan. We also establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify the role of Bier spheres in the theory of generalized permutohedra.
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