2016/05/02 by Bernardo González Merino, Merino, Bernardo González, Matthias Henze +1 · 2 citations
Mathematics · #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1605.00443
In 1978, Makai Jr. established a remarkable connection between the\nvolume-product of a convex body, its maximal lattice packing density and the\nminimal density of a lattice arrangement of its polar body intersecting every\naffine hyperplane. Consequently, he formulated a conjecture that can be seen as\na dual analog of Minkowski's fundamental theorem, and which is strongly linked\nto the well-known Mahler-conjecture.\n Based on the covering minima of Kannan & Lov 'asz and a problem posed by\nFejes T 'oth, we arrange Makai Jr.'s conjecture into a wider context and\ninvestigate densities of lattice arrangements of convex bodies intersecting\nevery i-dimensional affine subspace. Then it becomes natural also to formulate\nand study a dual analog to Minkowski's second fundamental theorem. As our main\nresults, we derive meaningful asymptotic lower bounds for the densities of such\narrangements, and furthermore, we solve the problems exactly for the special,\nyet important, class of unconditional convex bodies.\n