Geodesic
The Empty Sphere Part II
Canadian journal of mathematics, Tome 40 (1988) no. 5, pp. 1058-1073

Voir la notice de l'article provenant de la source Cambridge University Press

Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of integer solutions of where f satisfies the following condition (in which Z denotes the integers): 
Ryshkov, S. S.; Erdahl, R. M. The Empty Sphere Part II. Canadian journal of mathematics, Tome 40 (1988) no. 5, pp. 1058-1073. doi: 10.4153/CJM-1988-043-5
@article{10_4153_CJM_1988_043_5,
     author = {Ryshkov, S. S. and Erdahl, R. M.},
     title = {The {Empty} {Sphere} {Part} {II}},
     journal = {Canadian journal of mathematics},
     pages = {1058--1073},
     year = {1988},
     volume = {40},
     number = {5},
     doi = {10.4153/CJM-1988-043-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-043-5/}
}
TY  - JOUR
AU  - Ryshkov, S. S.
AU  - Erdahl, R. M.
TI  - The Empty Sphere Part II
JO  - Canadian journal of mathematics
PY  - 1988
SP  - 1058
EP  - 1073
VL  - 40
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-043-5/
DO  - 10.4153/CJM-1988-043-5
ID  - 10_4153_CJM_1988_043_5
ER  - 
%0 Journal Article
%A Ryshkov, S. S.
%A Erdahl, R. M.
%T The Empty Sphere Part II
%J Canadian journal of mathematics
%D 1988
%P 1058-1073
%V 40
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-043-5/
%R 10.4153/CJM-1988-043-5
%F 10_4153_CJM_1988_043_5

[1] 1. Coxeter, H. S. M., Regular poly topes (Dover, New York, 1973). Google Scholar

[2] 2. Coxeter, H. S. M., Regular and semi-regular poly topes. II, Mat. Z. 188 (1986), 559–591. Google Scholar

[3] 3. Delaunay, B., Sur la sphere vide, Proc. Internat. Congr. Math. I (Univ. of Toronto Press, Toronto, 1928), 695–700. Google Scholar

[4] 4. Delone, B. N., The geometry of positive quadratic forms, Uspehi Mat. Nauk 3, (1937) 16–62; 4, (1938) 102–164, (Russian). Google Scholar

[5] 5. Erdahl, R. M., A cone of inhomogeneous second order polynomials, to appear. Google Scholar

[6] 6. Erdahl, R. M. and Ryshkov, S. S., The empty sphere, Can. J. Math. 39 (1987), 794–824. Google Scholar

[7] 7. Ryshkov, S. S. and Erdahl, R. M., The geometry of the integer roots of some quadraticequations with many variables, Soviet Math. Dokl. 26 (1982). Google Scholar

[8] 8. Ryshkov, S. S. and Baranovskii, E. P., Classical methods in the theory of lattice packings, Uspehi Mat. Nauk 34 (1979); English transi, in Russian Math. Surveys 34 (1979). Google Scholar

[9] 9. Ryshkov, S. S. and Baranovskii, E. P., The C-types of n-dimensional lattices and the five-dimensional primitive par allelohedrons (with applications to the theory of covering), Trudy Mat. Inst. Steklov 137 (1976); Engl, transi. Proc. Steklov Inst. Mat. 137 (1976). Google Scholar

[10] 10. Ryshkov, S. S. and Šušbaev, S. Š., The structure of the L-partition for the second perfect lattice, Mat. Sbornik 116 (1981); English transi, in Math. USSR Sbornik 44 (1983). Google Scholar

[11] 11. Vornoi, G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire, J. Reine Angew. Math. 133 (1908), 79–178. Google Scholar

[12] 12. Vornoi, G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire, J. Reine Angew. Math. 134 (1908), 198–287; 136 (1909), 67–178. Google Scholar

Cité par Sources :