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arXiv · 2108.00618

Bier spheres of extremal volume and generalized permutohedra

Filip D. Jevtić·Rade T. Živaljević
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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.

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Filip D. Jevtić, Rade T. Živaljević. 2021-08-02. Bier spheres of extremal volume and generalized permutohedra. https://arxiv.org/abs/2108.00618

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