Virtual polytopes

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set [n] = {1, . . . , n}. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set [n + 1]. The cyclopermutohedron was introduced by the second author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (a) the volume of the cyclopermutohedron equals zero, and (b) the homology groups H k for k = 0, . . . , n −2 of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.

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Cyclopermutohedron: geometry and topology

Nekrasov

Panina

Zhukova

2016

European Journal of Mathematics

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Inscribable fans I: Inscribed cones and virtual polytopes

Manecke,

Sanyal

2024

Mathematika

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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 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 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 nontrivial 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.

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Towards Lower Bounds on the Depth of ReLU Neural Networks

Hertrich

Basu

Summa

et al. 2023

SIAM J. Discrete Math.

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Virtual polytopes

Cited by 3 publications

References 37 publications

Cyclopermutohedron: geometry and topology

Cyclopermutohedron: geometry and topology

Inscribable fans I: Inscribed cones and virtual polytopes

Towards Lower Bounds on the Depth of ReLU Neural Networks

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