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Bier spheres of extremal volume and generalized permutohedra (2108.00618v1)
Published 2 Aug 2021 in math.CO and math.MG
Abstract: A Bier sphere $Bier(K) = K\ast_\Delta K\circ$, defined as the deleted join of a simplicial complex and its Alexander dual $K\circ$, is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to $K$ a natural fan $Fan(K)$, related to the Braid fan. Along the way we 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.