Packing minima of convex bodies
Mei Han, Martin Henk, Fei Xue
TL;DR
The paper develops the theory of packing minima $\rho_i(K,\Lambda)$ for convex bodies $K$ containing the origin and lattices $\Lambda$, providing a sharp lower bound $\frac{n+1}{n!}\prod_{i=1}^n \frac{1}{\rho_i(K,\Lambda)} \le \frac{\mathrm{vol}(K)}{\det(\Lambda)}$ and a Davenport-type upper bound $\frac{\mathrm{vol}(K)}{\det(\Lambda)} \le \delta^l(K)\cdot\prod_{i=1}^n \frac{1}{\rho_i(K_s,\Lambda)}$, thereby interpolating between successive minima and those of the polar body. A projection-based framework and polar-transference bounds relate packing minima to both subspace projections and the polar lattice, with $\rho_n(K,\Lambda)=\lambda_1(K,\Lambda)$ and $\rho_i(K,\Lambda)$ bounded by $\lambda_{n-i+1}(K,\Lambda)$ and $1/\lambda_i(K^*,\Lambda^*)$ up to a factor $\gamma$. The authors compute exact minima for several canonical bodies, including the unit cube $C_n$, the cross-polytope $C_n^*$, and the simplex family $T_n$, and identify a complete characterization in the diagonal-formed class $\mathcal{K}^n_{\mathcal{A}}$, elucidating the interplay between geometry and lattice structure. Overall, the results advance understanding of lattice packing via packing minima and provide concrete, computable bounds for important convex bodies.
Abstract
In 2021, Henk, Schymura and Xue introduced packing minima, associated with a convex body and a lattice, as packing counterparts to the covering minima of Kannan and Lovász. Motivated by conjectures on the volume inequalities for the successive minima, we generalized the definition of the packing minima to the class of all convex bodies that contain the origin in their interior. For these packing minima, we presented several novel volume inequalities and calculated the specific values of the packing minima for several special convex bodies.