A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

  • Tianran ChenEmail author
  • Robert Davis
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 277)


The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full-dimensional polytopes is precisely the product of the normalized volumes of the summands.


Lattice polytopes Free sum BKK bound 


  1. 1.
    Alilooee, A., Soprunov, I., Validashti, J.: Generalized multiplicities of edge ideals. J. Algebraic Comb. 1–32 (2016)Google Scholar
  2. 2.
    Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3, 493–535 (1994)Google Scholar
  3. 3.
    Beck, M., Jayawant, P., McAllister, T.B.: Lattice-point generating functions for free sums of convex sets. J. Comb. Theory Ser. A 120, 1246–1262 (2013)., Scholar
  4. 4.
    Beck, M., Robins, S.: Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Springer, New York (2007)Google Scholar
  5. 5.
    Bernshtein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9, 183–185 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bihan, F., Soprunov, I.: Criteria for strict monotonicity of the mixed volume of convex polytopes (2017). arXiv:1702.07676 [math],
  7. 7.
    Braun, B.: An Ehrhart series formula for reflexive polytopes. Electron. J. Combin. 13(15), 5 pp. (electronic) (2006)Google Scholar
  8. 8.
    Braun, B.: Unimodality Problems in Ehrhart Theory, pp. 687–711. Springer International Publishing, Cham (2016). Scholar
  9. 9.
    Braun, B., Davis, R., Solus, L.: Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices (2016).
  10. 10.
    Chen, T.: Unmixing the mixed volume computation (2017). arXiv:1703.01684 [math],
  11. 11.
    Conrads, H.: Weighted projective spaces and reflexive simplices. Manuscripta Math. 107, 215–227 (2002). Scholar
  12. 12.
    Ehrhart, E.: Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)Google Scholar
  13. 13.
    Haase, C., Melnikov, I.V.: The reflexive dimension of a lattice polytope. Ann. Comb. 10, 211–217 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hartshorne, R.: Algebraic Geometry, no. 52. Springer (1977)CrossRefGoogle Scholar
  15. 15.
    Hibi, T.: Ehrhart polynomials of convex polytopes, \(h\)-vectors of simplicial complexes, and nonsingular projective toric varieties. In: Discrete and Computational Geometry: Papers from the DIMACS Special Year, vol. 6, pp. 165–177 (1991)Google Scholar
  16. 16.
    Hibi, T.: Algebraic Combinatorics on Convex Polytopes. Carslaw Publications, Australia (1992)zbMATHGoogle Scholar
  17. 17.
    Khovanskii, A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12, 38–46 (1978). Scholar
  18. 18.
    Kreuzer, M., Skarke, H.: Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2, 853–871 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kushnirenko, A.G.: A Newton polyhedron and the number of solutions of a system of k equations in k unknowns. Usp. Math. Nauk 30, 266–267 (1975)Google Scholar
  20. 20.
    Kushnirenko, A.G.: Newton polytopes and the Bezout theorem. Funct. Anal. Appl. 10, 233–235 (1976)., Scholar
  21. 21.
    McAllister, T.: Private Communication (2018)Google Scholar
  22. 22.
    Minkowski, H.: Theorie der konvexen Korper, insbesondere Begrundung ihres Oberflachenbegriffs. Gesammelte Abhandlungen von Hermann Minkowski 2, 131–229 (1911)Google Scholar
  23. 23.
    Payne, S.: Ehrhart series and lattice triangulations. Discrete Comput. Geom. 40, 365–376 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980); Combinatorial mathematics, optimal designs and their applications. In: Proceedings of the Symposium of Combinatorial Mathematics, Optimal Designs, Colorado State University, Fort Collins, Colorado (1978)Google Scholar
  25. 25.
    Stapledon, A.: Counting lattice points in free sums of polytopes. J. Comb. Theory Ser. A (2017). To appearGoogle Scholar
  26. 26.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence, RI (1996)Google Scholar
  27. 27.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer-Verlag, New York (1995)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAuburn University MontgomeryMontgomeryUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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