Vol. 10, No. 1, 2021

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Packing minima and lattice points in convex bodies

Martin Henk, Matthias Schymura and Fei Xue

Vol. 10 (2021), No. 1, 25–48
Abstract

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 packing minima, associated to a convex body K and a lattice Λ. 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 & Lovász (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.

Keywords
lattices, convex bodies, packing minima, successive minima, covering minima
Mathematical Subject Classification
Primary: 52C07
Secondary: 11H06, 52C05
Milestones
Received: 8 May 2020
Revised: 21 July 2020
Accepted: 7 August 2020
Published: 16 January 2021
Authors
Martin Henk
Institut für Mathematik
Technische Universität
Berlin
Germany
Matthias Schymura
Institut für Mathematik
BTU Cottbus-Senftenberg
Cottbus
Germany
Fei Xue
Institute of Mathematics
School of Mathematical Sciences
Nanjing Normal University
Nanjing, 210023
China