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Oleg N. German
Klein polyhedra and lattices with positive norm minima
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 175-190, doi: 10.5802/jtnb.580
Article PDF | Reviews MR 2332060 | Zbl pre05186981

Résumé - Abstract

A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of $\mathbb{R}^n$. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved that the facets of all the $2^n$ Klein polyhedra generated by a lattice $\Lambda $ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by $\Lambda $ and related to the positive orthant have uniformly bounded determinants.


[1] O. N. German, Klein polyhedra and norm minima of lattices. Doklady Mathematics 406:3 (2006), 38–41.  MR 2258502
[2] P. Erdös, P. Gruber, J. Hammer, Lattice Points. Pitman Monographs and Surveys in Pure and Applied Mathematics 39. Longman Scientific & Technical, Harlow (1989).  MR 1003606 |  Zbl 0683.10025
[3] F. Klein, Uber eine geometrische Auffassung der gewohnlichen Kettenbruchentwichlung. Nachr. Ges. Wiss. Gottingen 3 (1895), 357–359. Article |  JFM 26.0229.02
[4] O. N. German, Sails and norm minima of lattices. Mat. Sb. 196:3 (2005), 31–60; English transl., Russian Acad. Sci. Sb. Math. 196:3 (2005), 337–367.  MR 2144275 |  Zbl 1084.11035
[5] J.–O. Moussafir, Convex hulls of integral points. Zapiski nauch. sem. POMI 256 (2000).  Zbl 1025.52006
[6] V. I. Arnold, Continued fractions. Moscow: Moscow Center of Continuous Mathematical Education (2002).
[7] V. I. Arnold, Preface. Amer. Math. Soc. Transl. 197:2 (1999), ix–xii.
[8] E. I. Korkina, Two–dimensional continued fractions. The simplest examples. Proc. Steklov Math. Inst. RAS 209 (1995), 143–166.  MR 1422222 |  Zbl 0883.11034
[9] T. Bonnesen, W. Fenchel, Theorie der konvexen Körper. Berlin: Springer (1934).  MR 344997 |  Zbl 0008.07708
[10] B. Grünbaum, Convex polytopes. London, New York, Sydney: Interscience Publ. (1967).  MR 226496 |  Zbl 0163.16603
[11] P. McMullen, G. C. Shephard, Convex polytopes and the upper bound conjecture. Cambridge (GB): Cambridge University Press (1971).  MR 301635 |  Zbl 0217.46702
[12] G. Ewald, Combinatorial convexity and algebraic geometry. Sringer–Verlag New York, Inc. (1996).  MR 1418400 |  Zbl 0869.52001
[13] Z. I. Borevich, I. R. Shafarevich, Number theory. NY Academic Press (1966).  MR 195803 |  Zbl 0145.04902
[14] J. W. S. Cassels, H. P. F. Swinnerton–Dyer, On the product of three homogeneous linear forms and indefinite ternary quadratic forms. Phil. Trans. Royal Soc. London A 248 (1955), 73–96.  MR 70653 |  Zbl 0065.27905
[15] B. F. Skubenko, Minima of a decomposable cubic form of three variables. Zapiski nauch. sem. LOMI 168 (1988).  Zbl 0718.11026
[16] B. F. Skubenko, Minima of decomposable forms of degree $n$ of $n$ variables for $n\ge 3$. Zapiski nauch. sem. LOMI 183 (1990).  Zbl 0784.11028
[17] G. Lachaud, Voiles et Polyèdres de Klein. Act. Sci. Ind., Hermann (2002).
[18] L. Danzer, B. Grünbaum, V. Klee, Helly’s Theorem and its relatives. in Convexity (Proc. Symp. Pure Math. 7) 101–180, AMS, Providence, Rhode Island, 1963.  Zbl 0132.17401