65 research outputs found

    The boundary volume of a lattice polytope

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    For a d-dimensional convex lattice polytope P, a formula for the boundary volume is derived in terms of the number of boundary lattice points on the first \floor{d/2} dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.Comment: 21 pages; subsumes arXiv:1002.1908 [math.CO]; to appear in the Bulletin of the Australian Mathematical Societ

    A note on palindromic δ\delta-vectors for certain rational polytopes

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    Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ\delta-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.Comment: 4 page

    Toric Fano three-folds with terminal singularities

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    This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point

    Mutations of fake weighted projective planes

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    In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterisation of such mutations in terms of T-singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking-Prokhorov

    A note on palindromic δ-vectors for certain rational polytopes

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    Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact

    Fano polytopes

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    Fano polytopes are the convex-geometric objects corresponding to toric Fano varieties. We give a brief survey of classification results for different classes of Fano polytopes

    A note on palindromic δ-vectors for certain rational polytopes

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    Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact

    Seven new champion linear codes

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    We exhibit seven linear codes exceeding the current best known minimum distance d for their dimension k and block length n. Each code is defined over F₈, and their invariants [n,k,d] are given by [49,13,27], [49,14,26], [49,16,24], [49,17,23], [49,19,21], [49,25,16] and [49,26,15]. Our method includes an exhaustive search of all monomial evaluation codes generated by points in the [0,5] x [0,5] lattice square

    Reflexive polytopes of higher index and the number 12

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    We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexive polygons up to index 200. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number 12" property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances. The number 12 property also holds more generally for l-reflexive non-convex or self-intersecting polygonal loops. We conclude by discussing higher-dimensional examples and open questions

    A note on palindromic δ-vectors for certain rational polytopes

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    Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact
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