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

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