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arXiv · 2005.02234

Packing minima and lattice points in convex bodies

Martin Henk·Matthias Schymura·Fei Xue
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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 \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.

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Martin Henk, Matthias Schymura, Fei Xue. 2020-05-05. Packing minima and lattice points in convex bodies. https://doi.org/10.2140/moscow.2021.10.25

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