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 Funktsional. Anal. i Prilozhen., 2000, Volume 34, Issue 2, Pages 43–49 (Mi faa294)

Sails and Hilbert Bases

J.-O. Moussafir

Université Paris-Dauphine

Abstract: A Klein polyhedron is the convex hull of the nonzero integral points of a simplicial cone $C\subset\mathbb{R}^n$. There are relationships between these polyhedra and the Hilbert bases of monoids of integral points contained in a simplicial cone.
In the two-dimensional case, the set of integral points lying on the boundary of a Klein polyhedron contains a Hilbert base of the corresponding monoid. In general, this is not the case if the dimension is greater than or equal to three. However, in the three-dimensional case, we give a characterization of the polyhedra that still have this property. We give an example of such a sail and show that our criterion does not hold if the dimension is four.

DOI: https://doi.org/10.4213/faa294

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English version:
Functional Analysis and Its Applications, 2000, 34:2, 114–118

Bibliographic databases:

UDC: 512.7+514.17

Citation: J. Moussafir, “Sails and Hilbert Bases”, Funktsional. Anal. i Prilozhen., 34:2 (2000), 43–49; Funct. Anal. Appl., 34:2 (2000), 114–118

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa294
• https://doi.org/10.4213/faa294
• http://mi.mathnet.ru/eng/faa/v34/i2/p43

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. N. German, “Sails and Hilbert Bases”, Proc. Steklov Inst. Math., 239 (2002), 88–95
2. O. N. Karpenkov, “On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities”, Funct. Anal. Appl., 38:2 (2004), 102–110
3. O. N. German, “Sails and norm minima of lattices”, Sb. Math., 196:3 (2005), 337–365
4. O. N. Karpenkov, “On an Invariant Möbius Measure and the Gauss–Kuzmin Face Distribution”, Proc. Steklov Inst. Math., 258 (2007), 74–86
5. Karpenkov, ON, “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatshefte fur Mathematik, 152:3 (2007), 217
6. Karpenkov, ON, “CONSTRUCTING MULTIDIMENSIONAL PERIODIC CONTINUED FRACTIONS IN THE SENSE OF Klein”, Mathematics of Computation, 78:267 (2009), 1687
7. I. A. Makarov, “Interior Klein Polyhedra”, Math. Notes, 95:6 (2014), 795–805
8. A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
9. Susse T., “Stable Commutator Length in Amalgamated Free Products”, J. Topol. Anal., 7:4 (2015), 693–717
10. S. I. Veselov, “O tselykh tochkakh poliedrov dvukh tipov”, Zhurnal SVMO, 19:3 (2017), 24–30
11. A. A. Illarionov, “Distribution of facets of higher-dimensional Klein polyhedra”, Sb. Math., 209:1 (2018), 56–70
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