Bier spheres of extremal volume and generalized permutohedra

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.