Bier spheres of extremal volume and generalized permutohedra

TL;DR

This paper explores the geometric properties of Bier spheres, including their volume, polytopality, and connections to generalized permutohedra, revealing new insights into their structure and applications.

Contribution

It introduces geometric realizations of Bier spheres, computes their volume, provides criteria for polytopality, and links them to generalized permutohedra and the Braid fan.

Findings

Computed volumes of Bier spheres.

Established criteria for polytopality.

Connected Bier spheres to 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.