论文标题
在凸形机构中包装最小值和晶格点
Packing minima and lattice points in convex bodies
论文作者
论文摘要
由长期以来对经典不平等数字的离散化的猜想，我们研究了一组新的参数，我们称之为\ emph {packing minima}，与凸面$ k $和lattice $ $λ$相关。这些数字在$ k $的连续最小值与$ k $的极性最小值的倒数之间插值，并且可以理解为与Kannan＆Lovász（1988）的覆盖最小值的包装相对。作为我们的主要结果，我们证明了将$ k $中的体积和晶格点的数量与包装最小值的顺序相关的急剧不平等。此外，我们扩展了经典的转移范围，并详细讨论了自然的示例类。
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 $Λ$. 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.
