The Reflexive Dimension of a Lattice Polytope

“…As special as reflexive polytopes are, it is interesting to note that in some sense they are rather general [19]: Proposition 2.2. Any lattice polytope is isomorphic to the face of a reflexive polytope.…”

Section: Reflexive Polytopesmentioning

confidence: 99%

Fano polytopes

Kasprzyk

Nill

2012

Strings, Gauge Fields, and the Geometry Behind

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“…It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. This fact naturally lead us to the study of the following question: Question Is every normal polytope unimodularly equivalent to a face of some normal Gorenstein Fano polytope?…”

Section: Introductionmentioning

confidence: 99%

Facets and volume of Gorenstein Fano polytopes

Hibi

Tsuchiya

2017

Mathematische Nachrichten

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It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension d+1. Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.

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“…Reflexive polytopes form an interesting class of lattice polytopes that are being studied for various reasons [4,10,17] and whose number grows very fast with increasing dimension [13,14]. On the other hand, the purely combinatorial notion of quivers furnishes us with one of the very rare systematic constructions of reflexive polytopes, called flow polytopes, also in high dimensions [1,2,11].…”

Section: Introductionmentioning

confidence: 99%

Flow polytopes and the graph of reflexive polytopes

Altmann

Nill

Schwentner

et al. 2009

Discrete Mathematics

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a b s t r a c tWe suggest defining the structure of an unoriented graph R d on the set of reflexive polytopes of a fixed dimension d. The edges are induced by easy mutations of the polytopes to create the possibility of walks along connected components inside this graph. For this, we consider two types of mutations: Those provided by performing duality via nef-partitions, and those arising from varying the lattice. Then for d ≤ 3, we identify the flow polytopes among the reflexive polytopes of each single component of the graph R d . For this, we present for any dimension d ≥ 2 an explicit finite list of quivers giving all d-dimensional reflexive flow polytopes up to lattice isomorphism. We deduce as an application that any such polytope has at most 6(d − 1) facets.

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The Reflexive Dimension of a Lattice Polytope

Abstract: The reflexive dimension refldim(P) of a lattice polytope P is the minimal integer d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k.

Cited by 24 publications

References 12 publications

Fano polytopes

Fano polytopes

Facets and volume of Gorenstein Fano polytopes

Flow polytopes and the graph of reflexive polytopes

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