Packing minima and lattice points in convex bodies
| Autor: | Matthias Schymura, Fei Xue, Martin Henk |
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| Rok vydání: | 2020 |
| Předmět: |
Sequence
Algebra and Number Theory Discretization Geometry of numbers Regular polygon Lattice (group) packing minima Inverse Metric Geometry (math.MG) successive minima Combinatorics Maxima and minima 52C05 Mathematics - Metric Geometry lattices 52C07 FOS: Mathematics Discrete Mathematics and Combinatorics Convex body Mathematics - Combinatorics convex bodies Combinatorics (math.CO) covering minima 11H06 Mathematics |
| Zdroj: | Mosc. J. Comb. Number Theory 10, no. 1 (2021), 25-48 |
| DOI: | 10.48550/arxiv.2005.02234 |
| Popis: | Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $\Lambda$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lov\'{a}sz (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail. Comment: 23 pages |
| Databáze: | OpenAIRE |
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