A Blichfeldt-type inequality for centrally symmetric convex bodies
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A Blichfeldt-type inequality for centrally symmetric convex bodies Matthias Henze
Received: 23 January 2012 / Accepted: 11 December 2012 / Published online: 21 December 2012 © Springer-Verlag Wien 2012
Abstract In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of a centrally symmetric convex body. Our main tool is a generalization of a result of Davenport that bounds the number of lattice points in terms of volumes of suitable projections. Keywords Blichfeldt-type inequalities · Davenport inequality · Central symmetry · Lattice point enumerator · Lattice points · Volume inequalities Mathematics Subject Classification (2000)
52C07 · 52B20 · 52A40 · 11H06
1 Introduction Let Kn be the set of all convex bodies in Rn , i.e., compact convex sets K with nonempty interior intK . Such a body K is called centrally symmetric if K = −K . The family of n-dimensional lattices in Rn is denoted by Ln and the usual Lebesgue measure with respect to the n-dimensional Euclidean space by voln (·). If the ambient space is clear from the context we omit the subscript and just write vol(·). For a given bounded subset S ⊂ Rn , the number of lattice points in S is denoted by G(S) = #(S ∩ Zn ). A lattice polytope is a polytope all of whose vertices are points in Zn . Finally, for an A ⊆ Rn the dimension of its affine hull will be denoted by dim A.
Communicated by A. Constantin. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the project He 2272/4-1. M. Henze (B) Fachbereich für Mathematik und Informatik, Freie Universität Berlin, Takustraße 9, 10823 Berlin, Germany e-mail: [email protected]
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We are interested in bounds on the volume in terms of the lattice point enumerator of a convex body. For K ∈ Kn with dim(K ∩ Zn ) = n a sharp lower bound on vol(K ) was obtained by Blichfeldt [4]. It reads vol(K ) ≥
1 (G(K ) − n) . n!
We will call results of this kind Blichfeldt-type inequalities. On the other hand, the best known upper bound on vol(P) for a lattice polytope P ∈ Kn is due to Pikhurko [15] vol(P) ≤ (8n)n 15n2
2n+1
G(intP)
and holds under the condition that G(intP) = 0. On the class of centrally symmetric convex bodies, Blichfeldt [4] and van der Corput [6] obtained a sharp upper bound on the volume vol(K ) ≤ 2n−1 (G(intK ) + 1) ,
K ∈ K0n .
(1.1)
Bey, Henk and Wills [3] proposed the study of a reverse inequality also on the class of centrally symmetric convex bodies, and in [11] the authors derive, as a first step, Blichfeldt-type inequalities for lattice crosspolytopes, lattice zonotopes, and for centrally symmetric planar convex sets. Moreover, they conjecture that n there is a constant c > 1 such that vol(K ) ≥ cn! G(K ), for every K ∈ K0n with dim(K ∩ Zn ) = n. In this work, we confirm this conjecture asymptotically by showing that for every ε ∈ (0, 1] and large enough n ∈ N a valid choice for this constant is c = 2 − ε. As the main ingredient to our argument we prove the following generalization of a
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