On densities of lattice arrangements intersecting every i-dimensional affine subspace

TL;DR

This paper explores the densities of lattice arrangements intersecting all affine subspaces, extending Makai Jr.'s conjecture and deriving bounds, with exact solutions for unconditional convex bodies.

Contribution

It generalizes Makai Jr.'s conjecture to i-dimensional affine subspaces and provides asymptotic bounds, including exact results for unconditional convex bodies.

Findings

Derived asymptotic lower bounds for lattice arrangement densities.

Solved the problem exactly for unconditional convex bodies.

Connected the problem to dual analogs of Minkowski's theorems.

Abstract

In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski's fundamental theorem, and which is strongly linked to the well-known Mahler-conjecture. Based on the covering minima of Kannan & Lov\'asz and a problem posed by Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and investigate densities of lattice arrangements of convex bodies intersecting every i-dimensional affine subspace. Then it becomes natural also to formulate and study a dual analog to Minkowski's second fundamental theorem. As our main results, we derive meaningful asymptotic lower bounds for the densities of such arrangements, and furthermore, we solve the problems exactly for the special, yet important, class of unconditional convex bodies.