Geodesic
Klein polyhedra and relative minima of lattices
Matematičeskie zametki, Tome 79 (2006) no. 4, pp. 546-552.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that in $\mathbb R^3$, the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron. We also prove, for almost any lattice in $\mathbb R^3$, that the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron. 
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O. N. German. Klein polyhedra and relative minima of lattices. Matematičeskie zametki, Tome 79 (2006) no. 4, pp. 546-552. http://geodesic.mathdoc.fr/item/MZM_2006_79_4_a4/

[1] Klein F., “Über eine geometrische Auffassung der gewohnlichen Kettenbruchentwichlung”, Nachr. Ges. Wiss. Göttingen, 3 (1895), 357–359

[2] Voronoi G. F., Sobranie sochinenii, T. 1, Izd-vo AN USSR, Kiev, 1952

[3] Minkowski H., “Généralisation de la théorie des fractions continues”, Ann. Sci. École Norm. Sup. (3), 13:2 (1896), 41–60 | MR | Zbl

[4] Bykovskii V. A., “Otnositelnye minimumy reshetok i vershiny mnogogrannikov Kleina”, Funktsion. analiz i ego prilozh., 40:1 (2006), 69–71 | MR | Zbl

[5] Moussafir J.-O., “Convex hulls of integral points”, Zapiski nauch. sem. POMI, 266, POMI, SPb., 2000, 188–217

[6] Dantser L., Gryunbaum B., Kli V., Teorema Khelli i ee primeneniya, Mir, M., 1968

[7] Steinitz E., “Bedingt konvergente Reihen und konvexe Systeme. I; II; III”, J. Reine Angew. Math., 143 (1913), 128–175 ; 144 (1914), 1–40 ; 146 (1916), 1–52 | Zbl | Zbl

[8] German O. N., “Parusa i bazisy Gilberta”, Tr. MIAN, 239, Nauka, M., 2002, 98–105 | MR | Zbl

[9] White G. K., “Lattice tetrahedra”, Canad. J. Math., 16 (1964), 389–396 | MR | Zbl