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How to stab a polytope

Sebastian Seemann, Francesca Zaffalon

TL;DR

This work develops a Schubert-arrangements framework to characterize the set of $k$-stabbing subspaces $P^{[k]}$ of a polytope $P$ in $\mathbb{P}^{n-1}$ by embedding the problem in the Grassmannian $\mathrm{Gr}(k,n)$ and using Chow forms of the faces. A chamber decomposition into stabbing chambers is obtained, with each chamber defined by fixed sign patterns of Chow forms restricted to relevant faces; $P^{[k]}$ is recovered as the closure of the maximally stabbing set $P^{[k]}_{\max}$ via semialgebraic inequalities built from these signs. The paper provides explicit criteria for simplicial polytopes and connects the stabbing construction to amplituhedra and loop geometries, offering a bridge between polyhedral geometry, Schubert calculus, and positive geometries. It also outlines open problems on the topology of polytopal Schubert arrangements and potential canonical forms in this setting, with implications for both theory and computation.

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.

How to stab a polytope

TL;DR

This work develops a Schubert-arrangements framework to characterize the set of -stabbing subspaces of a polytope in by embedding the problem in the Grassmannian and using Chow forms of the faces. A chamber decomposition into stabbing chambers is obtained, with each chamber defined by fixed sign patterns of Chow forms restricted to relevant faces; is recovered as the closure of the maximally stabbing set via semialgebraic inequalities built from these signs. The paper provides explicit criteria for simplicial polytopes and connects the stabbing construction to amplituhedra and loop geometries, offering a bridge between polyhedral geometry, Schubert calculus, and positive geometries. It also outlines open problems on the topology of polytopal Schubert arrangements and potential canonical forms in this setting, with implications for both theory and computation.

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.

Paper Structure

This paper contains 12 sections, 1 theorem, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Schubert arrangements and Chow forms
  3. Grassmannians and Schubert divisors
  4. The Schubert arrangement of a polytope
  5. Loop amplituhedra and stabbed polytopes
  6. Stabbing sets and Chow forms
  7. Chamber decomposition of $P^{[k]}_{\max}$
  8. Recovering $P^{[k]}$ from $P^{[k]}_{\max}$
  9. Equations defining $P^{[k]}$
  10. How to stab a simplicial polytope
  11. Amplituhedra and stabbing sets
  12. Open problems

Key Result

lemma 1

For $k,n,P$ fixed as above, the vector of Chow forms $\mathrm{C}_P^k(V) \in \mathbb{P}^{f-1}$ for every $V\in \mathop{\mathrm{Gr}}\nolimits(k,n)$, where $f$ is the number of $(n-k-1)$-dimensional faces of $P$.

Figures (2)

  • Figure 1: A visual representation of Example \ref{['ex : bad example']}. We can see that the line $V$ stabs the simplex and the line $W$ doesn't. Moreover we can see the points that the path constructed in the proof of Theorem \ref{['thm : sign var']} would follow. These exit the polytope without crossing any divisor in the Schubert arrangement considered.
  • Figure 2: The line $V$ from Example \ref{['ex : octahedron']} stabbing the octahedron and the sign condition on the boundaries of the stabbed faces.

Theorems & Definitions (11)

  • lemma 1
  • proof
  • Example 3
  • proof
  • proof
  • Example 4: Schubertarr
  • proof
  • proof
  • proof
  • proof
  • ...and 1 more