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Inscribable fans I: Inscribed cones and virtual polytopes

Sebastian Manecke, Raman Sanyal

Abstract

We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope $P$. We show that the associated space of polytopes, called the inscribed cone of $P$, is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to $P$ is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension $2$ as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan $\mathcal{N}$ form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of $\mathcal{N}$. Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be non-trivial even if $\mathcal{N}$ does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.

Inscribable fans I: Inscribed cones and virtual polytopes

Abstract

We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope . We show that the associated space of polytopes, called the inscribed cone of , is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of . Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be non-trivial even if does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.

Paper Structure

This paper contains 24 sections, 63 theorems, 104 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Ideal hyperbolic type cones
  3. Deformations of Delaunay subdivisions
  4. Inscribed virtual polytopes
  5. Routed billiard trajectories and reflection groupoids
  6. Local reflections and inscribable fans
  7. Local reflections
  8. Virtually inscribable fans and the reflection game
  9. Inscribed Permutahedra
  10. Delaunay subdivisions and visibility complexes
  11. Inscribable fans in dimension $2$
  12. Inscribable fans in general dimensions
  13. Restrictions from faces
  14. Simple and even polytopes
  15. Inscribed Nestohedra
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $P$ be a convex $d$-dimensional polytope and $Q,Q' \in \mathcal{I}_+(P)$. Then In particular, $\mathcal{I}_+(P)$ has the structure of an open polyhedral cone of dimension $\le d$.

Figures (11)

  • Figure 1: Three normally equivalent hexagons. The rightmost one is inscribed.
  • Figure 2: Two positively co-circular Delaunay subdivisions.
  • Figure 3: A $2$-dimensional fan all whose inscribed polytopes are virtual.
  • Figure 4: Three positively co-circular segments in the plane.
  • Figure 5: Left: A $2$-dimensional fan $\mathcal{N}$ with regions and its profile $\beta = (\beta_0, \beta_1, \beta_2, \beta_3, \beta_4)$. Right: An inscribed polygon with normal fan $\mathcal{N}$ and labels as in the proof of Theorem \ref{['thm:2d_insc']}.
  • ...and 6 more figures

Theorems & Definitions (117)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • ...and 107 more