Abstract
For a convex bodyK⊂E 2 and a latticeL⊂E 2 let μ i (K, L),i=1, 2, denote its covering minima introduced by Kannan and Lovasz. We show μ1(K, L) μ2(K, L)V(K)≥3/4 det(L), whereV denotes the area. This inequality is tight and there are five different cases of equality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Betke, U., Henk, M. and Wills, J. M.: Successive-minima-type inequalities,Discrete Comput. Geom. 9 (1993), 165–175.
Bourgain, J. and Milman, V. D.: Sections euclidiennes et volume des corps symétriques convexes dans ℝn,Acad. Sci. Paris 300 (13) (1985), 435–438.
Cassels, J. W. S.:An Introduction to the Geometry of Numbers, Springer, Berlin, Heidelberg, New York, 1971.
Fejes, Tóth, G.: Research problem 18,Period. Math. Hungar. 7 (1976), 89–90.
Fejes Tóth, L. and Makai, E. Jr: On the thinnest non-separable lattice of convex plates,Studia Sci. Math. Hungar. 9 (1974), 191–193.
Gruber, P. M. and Lekkerkerker, C. G.:Geometry of Numbers, North-Holland, Amsterdam, 1987.
Hurkens, C. A. J.: Blowing up convex sets in the plane,Linear Algebra Appl. 134 (1990), 121–128.
Kannan, R. and Lovasz, L.: Covering minima and lattice-point-free convex bodies,Ann. Math. 128 (1988), 577–602.
Mahler, K.: Polar analogues of two theorems by Minkowski,Bull. Austr. Math. Soc. 11 (1974), 121–129.
Mahler, K.: Ein Übertragungsprinzip für konvexe Körper,Časopis Pěst. Mat. Fys. 68 (1939), 93–102.
Makai, E. Jr: On the thinnest non-separable lattice of convex bodies,Studia Sci. Math. Hungar. 13 (1978), 19–27.
Rogers, C. A. and Shephard, G. C.: The difference body of a convex body,Arch. Math. 8 (1957), 220–233.