Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T04:08:02.639Z Has data issue: false hasContentIssue false
  • English
  • Français

THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 September 2011

GÁBOR HEGEDÜS  and
ALEXANDER M. KASPRZYK
GÁBOR HEGEDÜS
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria (email: [email protected])
ALEXANDER M. KASPRZYK*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

Keywords

lattice polytopeboundary volumereflexive polytopeorder polytopeBirkhoff polytope

MSC classification

Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 52B12: Special polytopes (linear programming, centrally symmetric, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 84 - 104
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Batyrev, V. V., ‘On the classification of smooth projective toric varieties’, Tohoku Math. J. 43(4) (1991), 569–585.CrossRefGoogle Scholar
[2]Batyrev, V. V., ‘Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties’, J. Algebraic Geom. 3(3) (1994), 493–535.Google Scholar
[3]Batyrev, V. V., ‘On the classification of toric Fano 4-folds’, J. Math. Sci. (New York) 94(1) (1999), 1021–1050.CrossRefGoogle Scholar
[4]Beck, M., De Loera, J. A., Develin, M., Pfeifle, J. and Stanley, R. P., ‘Coefficients and roots of Ehrhart polynomials’, in: Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization, Contemporary Mathematics, 374 (American Mathematical Society, Providence, RI, 2005), pp. 15–36.CrossRefGoogle Scholar
[5]Beck, M. and Pixton, D., ‘The Ehrhart polynomial of the Birkhoff polytope’, Discrete Comput. Geom. 30(4) (2003), 623–637.CrossRefGoogle Scholar
[6]Beck, M. and Robins, S., Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, Undergraduate Texts in Mathematics (Springer, New York, 2007).Google Scholar
[7]Beck, M. and Sottile, F., ‘Irrational proofs for three theorems of Stanley’, European J. Combin. 28(1) (2007), 403–409.CrossRefGoogle Scholar
[8]Bremner, D. and Klee, V., ‘Inner diagonals of convex polytopes’, J. Combin. Theory Ser. A 87(1) (1999), 175–197.CrossRefGoogle Scholar
[9]Canfield, E. R. and McKay, B. D., ‘The asymptotic volume of the Birkhoff polytope’, Online J. Anal. Comb. (4) (2009), Art. 2, 4.Google Scholar
[10]Casagrande, C., ‘The number of vertices of a Fano polytope’, Ann. Inst. Fourier (Grenoble) 56(1) (2006), 121–130.CrossRefGoogle Scholar
[11]Danilov, V. I., ‘The geometry of toric varieties’, Uspekhi Mat. Nauk 33(2(200)) (1978), 85–134, 247.Google Scholar
[12]Diaz, R. and Robins, S., ‘The Ehrhart polynomial of a lattice polytope’, Ann. of Math. (2) 145(3) (1997), 503–518.CrossRefGoogle Scholar
[13]Ehrhart, E., ‘Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires’, J. Reine Angew. Math. 227 (1967), 25–49.Google Scholar
[14]Fiset, M. H. J. and Kasprzyk, A. M., ‘A note on palindromic δ-vectors for certain rational polytopes’, Electron. J. Combin. 15(N18).Google Scholar
[15]Haase, C. and Melnikov, I. V., ‘The reflexive dimension of a lattice polytope’, Ann. Comb. 10 (2006), 211–217.CrossRefGoogle Scholar
[16]Hibi, T., ‘Ehrhart polynomials of convex polytopes, h-vectors of simplicial complexes, and nonsingular projective toric varieties’, in: Discrete and Computational Geometry (New Brunswick, NJ, 1989/1990), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 6 (American Mathematical Society, Providence, RI, 1991), pp. 165–177.Google Scholar
[17]Hibi, T., ‘A lower bound theorem for Ehrhart polynomials of convex polytopes’, Adv. Math. 105(2) (1994), 162–165.CrossRefGoogle Scholar
[18]Kasprzyk, A. M., ‘Toric Fano varieties and convex polytopes’, PhD Thesis, University of Bath, Available from http://hdl.handle.net/10247/458.Google Scholar
[19]Klee, V., ‘A combinatorial analogue of Poincaré’s duality theorem’, Canad. J. Math. 16 (1964), 517–531.CrossRefGoogle Scholar
[20]Kołodziejczyk, K., ‘On the volume of lattice manifolds’, Bull. Aust. Math. Soc. 61(2) (2000), 313–318.CrossRefGoogle Scholar
[21]Macdonald, I. G., ‘The volume of a lattice polyhedron’, Proc. Cambridge Philos. Soc. 59 (1963), 719–726.CrossRefGoogle Scholar
[22]Macdonald, I. G., ‘Polynomials associated with finite cell-complexes’, J. Lond. Math. Soc. (2) 4 (1971), 181–192.CrossRefGoogle Scholar
[23]Øbro, M., ‘An algorithm for the classification of smooth Fano polytopes’, arXiv:0704.0049v1 [math.CO], Classifications available from http://grdb.lboro.ac.uk/.Google Scholar
[24]Pak, I., ‘Four questions on Birkhoff polytope’, Ann. Comb. 4(1) (2000), 83–90.CrossRefGoogle Scholar
[25]Park, H. S., ‘The f-vectors of some toric Fano varieties’, J. Appl. Math. Comput. 11(1–2) (2003), 437–444.Google Scholar
[26]Pick, G. A., ‘Geometrisches zur Zahlenlehre’, Sitzungber. Lotos 19 (1899), 311–319.Google Scholar
[27]Pommersheim, J. E., ‘Toric varieties, lattice points and Dedekind sums’, Math. Ann. 295(1) (1993), 1–24.CrossRefGoogle Scholar
[28]Reeve, J. E., ‘On the volume of lattice polyhedra’, Proc. Lond. Math. Soc. (3) 7 (1957), 378–395.CrossRefGoogle Scholar
[29]Stanley, R. P., ‘A chromatic-like polynomial for ordered sets’, in: Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (Univ. North Carolina, Chapel Hill, N.C, 1970), pp. 421–427.Google Scholar
[30]Stanley, R. P., ‘Hilbert functions of graded algebras’, Adv. Math. 28(1) (1978), 57–83.CrossRefGoogle Scholar
[31]Stanley, R. P., ‘Decompositions of rational convex polytopes’, Ann. Discrete Math. 6 (1980), 333–342; Combinatorial Mathematics, Optimal Designs and their Applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978).CrossRefGoogle Scholar
[32]Stanley, R. P., ‘Two poset polytopes’, Discrete Comput. Geom. 1(1) (1986), 9–23.CrossRefGoogle Scholar
[33]Stanley, R. P., Enumerative Combinatorics Vol. 1, Cambridge Studies in Advanced Mathematics, 49 (Cambridge University Press, Cambridge, 1997), with a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 original.CrossRefGoogle Scholar
[34]Turner, L. R., ‘Inverse of the Vandermonde matrix with applications’. NASA Technical Note.Google Scholar