How to stab a polytope

TL;DR

This paper investigates the geometric problem of characterizing linear subspaces intersecting a polytope, introducing a Schubert arrangement framework using Chow forms to describe the set of such subspaces.

Contribution

It introduces a novel Schubert arrangement approach using Chow forms to describe and analyze the set of subspaces intersecting a polytope.

Findings

Defined the set of intersecting subspaces via sign conditions on Chow forms

Provided inequalities characterizing stabbing subspaces

Connected polytope face intersections with Grassmannian geometry

Abstract

We study the set of linear subspaces of a fixed dimension intersecting a given polytope. To describe this set as a semialgebraic subset of a Grassmannian, we introduce a Schubert arrangement of the polytope, defined by the Chow forms of the polytope's faces of complementary dimension. We show that the set of subspaces intersecting a specified family of faces is defined by fixing the sign of the Chow forms of their boundaries. We give inequalities defining the set of stabbing subspaces in terms of sign conditions on the Chow form.