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Lattice inequalities for convex bodies and arbitrary lattices. (English) Zbl 0792.52008

For a convex body \(K\) in Euclidean \(E^ d\) let \(V(K)\) and \(F(K)\) denotes its volume and surface area. If \(L \subset E^ d\) is a lattice (with \(\text{det} L>0)\), then \(G(K,L)=\text{card} (K \cap L)\) denotes the lattice point enumerator. Further \(\lambda_ d(L)\) denotes the last of the classical successive minima of \(L\) with respect to the unit ball \(B^ d\). Then the author proves \[ G(K,L) \text{det} L \geq V(K)-{1 \over 2} \lambda_ d(L) F(K) \] which is tight for orthogonal lattice and generalizes a result by J. Bokowski, H. Hadwiger and the reviewer [Math. Z. 127, 363-364 (1972; Zbl 0238.52005)]. The author further proves for “lattice-periodic” sets: \[ \lambda_ d(L) F(A) \geq 8V(A)(1-V(A)/ \text{det} L) \] again generalizing a classical isoperimetric inequality by H. Hadwiger [Monatsh. Math. 76, 410-418 (1972; Zbl 0248.52012)].
Reviewer: J.M.Wills (Siegen)

MSC:

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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References:

[1] Bokowski, J., Hadwiger, H., Wills, J. M.: Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper imn-dimensionalen Raum. Math. Z.127, 363-364 (1972). · Zbl 0238.52005 · doi:10.1007/BF01111393
[2] Gruber, P. M., Lekkerkerker, C. G.: Geometry of Numbers. Amsterdam: North Holland. 1987. · Zbl 0611.10017
[3] Hadwiger, H.: Gitterperiodische Punktmengen und Isoperimetrie. Mh. Math.76, 410-418 (1972). · Zbl 0248.52012 · doi:10.1007/BF01297304
[4] Lagarias, J. C., Lenstra Jr, H. W., Schnorr, C. P.: Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica10, 333-348 (1990). · Zbl 0723.11029 · doi:10.1007/BF02128669
[5] Schnell, U.: Minimal determinants and lattice inequalities. Bull. London Math. Soc.24, 606-612 (1992). · Zbl 0780.52017 · doi:10.1112/blms/24.6.606
[6] Schnell, U., Wills, J. M.: Two isoperimetric inequalities with lattice constraints. Mh. Math.112, 227-233 (1991). · Zbl 0737.52008 · doi:10.1007/BF01297342
[7] Schnell, U., Wills, J. M.: On successive minima and intrinsic volumes. To appear in Mathematika. · Zbl 0781.52009
[8] Wills, J. M.: Bounds for the lattice point enumerator. Geom. Dedicata40, 237-244 (1991). · Zbl 0738.52018 · doi:10.1007/BF00145917
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