On the integer points in a lattice polytope: n-fold Minkowski sum and boundary

  • Marko LindnerEmail author
  • Steffen Roch
Original Paper


In this article we compare the set of integer points in the homothetic copy \({n\Pi}\) of a lattice polytope \({\Pi\subseteq{{\mathbb R}}^d}\) with the set of all sums x 1 + . . . + x n with \({x_1,\ldots,x_n\in \Pi\cap{{\mathbb Z}}^d}\) and \({n\in{{\mathbb N}}}\) . We give conditions on the polytope \({\Pi}\) under which these two sets coincide and we discuss two notions of boundary for subsets of \({{{\mathbb Z}}^d}\) or, more generally, subsets of a finitely generated discrete group.


Lattice polytopes Integer points Boundary Projection methods 

Mathematics Subject Classification (2000)

52B20 52C07 65J10 


  1. Adachi T.: A note on the Følner condition for amenability. Nagoya Math. J. 131, 67–74 (1993)MathSciNetzbMATHGoogle Scholar
  2. Awyong P.W., Henk M., Scott P.R.: Note on lattice-point-free convex bodies. Monatsh Math. 126, 7–12 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. Beck M., Robins S.: Computing the Continuous Discretely. Springer, Berlin (2007)zbMATHGoogle Scholar
  4. Betke V., Kneser M.: Zerlegungen und Bewertungen von Gitterpolytopen. J Reine Angew Math 358, 202–208 (1985)MathSciNetzbMATHGoogle Scholar
  5. Bokowski J., Hadwiger H., Wills J.M.: Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im n-dimensionalen euklidischen Raum. Math Z 127, 363–364 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Gritzmann, P., Wills, J.M.: Lattice points. In: Handbook of Convex Geometry vol. B, pp. 765–797, North-Holland, Amsterdam (1993)Google Scholar
  7. Gruber, P.M.: Convex and Discrete Geometry. Springer Series: Grundlehren der mathematischen Wissenschaften, vol. 336 (2007)Google Scholar
  8. Grünbaum B.: Convex Polytopes, 2nd edn. Graduate Texts in Mathematics, vol. 221. Springer, Berlin (2003)Google Scholar
  9. Henk, M., Richter-Gebert, J., Ziegler, G.M.: Basic properties of convex polytopes, In: Goodman J.E., O’Rourke J. (eds.) Handbook of Discrete and Computational Geometry (Chapter 14), Chapmann & Hall (2004).Google Scholar
  10. Kantor J.M.: Triangulations of integral polytopes and Ehrhart polynomials. Contributions Algebra Geom. 39(1), 205–218 (1998)MathSciNetzbMATHGoogle Scholar
  11. Kantor J.M.: On the width of lattice-free simplices. Compositio Mathematica 118, 235–241 (1991)MathSciNetCrossRefGoogle Scholar
  12. Kempf G., Knudsen F., Mumford D., Saint-Donat B.: Toroidal Embeddings 1. Lecture Notes 339. Springer, Berlin (2008)Google Scholar
  13. Lindner, M.: Fredholm Theory and Stable Approximation of Band Operators and Generalisations. Habilitation thesis, TU Chemnitz (2009)Google Scholar
  14. Lindner M.: Stable subsequences of the finite section method. J. Appl. Numer. Math. 60, 501–512 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory, Birkhäuser (2004)Google Scholar
  16. Reeve J.E.: On the volume of lattice polyhedra. Proc. London Math. Soc. 7, 378–395 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  17. Roch, S.: Finite sections of band-dominated operators. Memoirs of the AMS. Vol. 191, Nr. 895 (2008)Google Scholar
  18. Roch S.: Spatial discretization of Cuntz algebras. Houston J. Math. 36(4), 1097–1132 (2010)MathSciNetzbMATHGoogle Scholar
  19. Roch S.: Spatial discretization of restricted group C*-algebras. Oper. Matrices 5(1), 53–78 (2011)MathSciNetzbMATHGoogle Scholar
  20. Schrijver A.: Theory of Linear and Integer Programming. Wiley, Lomdon (1986)zbMATHGoogle Scholar
  21. White G.K.: Lattice polyhedra. Canad J. Math. 16, 389–396 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ziegler G.M.: Lectures on Polytopes. Graduate Texts in Mathematics 152. Springer, Berlin (1995)Google Scholar

Copyright information

© The Managing Editors 2011

Authors and Affiliations

  1. 1.Fakultät MathematikTU ChemnitzChemnitzGermany
  2. 2.TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Institut für Angewandte AnalysisFreibergGermany
  3. 3.Fachbereich MathematikTU DarmstadtDarmstadtGermany

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