×

Two isoperimetric inequalities with lattice constraints. (English) Zbl 0737.52008

Two isoperimetric inequalities with lattice constraints for arbitrary lattices in the euclidean plane are proved. We generalize previous results by J. Bokowski, H. Hadwiger, and the second author [Math. Z. 127, 363-364 (1972; Zbl 0238.52005)] for the integer lattice \(\mathbb{Z}^ d\) (but all dimensions \(d\)) to general lattices. For arbitrary \(d\) a partial result is given.

MSC:

52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry

Citations:

Zbl 0238.52005
PDFBibTeX XMLCite
Full Text: DOI EuDML

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] Wills, J. M.: Kugellagerungen und Konvexgeometrie. Jahresber. d. DMV92, 21-46 (1990). · Zbl 0688.52004
[5] Wills, J. M.: Bounds for the lattice point enumerator, in press. · Zbl 0738.52018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.