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Abstract
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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
and a
lattice
.
These numbers interpolate between the successive minima of
and the inverse of the successive minima of the polar body of
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
to the sequence of packing minima. Moreover, we extend classical transference
bounds and discuss a natural class of examples in detail.
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Keywords
lattices, convex bodies, packing minima, successive minima,
covering minima
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Mathematical Subject Classification
Primary: 52C07
Secondary: 11H06, 52C05
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Milestones
Received: 8 May 2020
Revised: 21 July 2020
Accepted: 7 August 2020
Published: 16 January 2021
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