Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In 1978, Makai Jr. established a remarkable connection beten 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 ll-known Mahler-conjecture. #R##N#Based on the covering minima of Kannan & Lov\'asz and a problem posed by Fejes T\'oth,  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,  derive meaningful asymptotic lor bounds for the densities of such arrangements, and furthermore,  solve the problems exactly for the special, yet important, class of unconditional convex bodies.
['Bernardo González Merino', 'Matthias Henze']
I am writing a research paper on a idea On densities of lattice arrangements intersecting every i-dimensional affine subspace
[{"from": "human", "value": "I am writing a research paper on a idea On densities of lattice arrangements intersecting every i-dimensional affine subspace"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In 1978, Makai Jr. established a remarkable connection beten 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 ll-known Mahler-conjecture. #R##N#Based on the covering minima of Kannan & Lov\\'asz and a problem posed by Fejes T\\'oth,  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,  derive meaningful asymptotic lor bounds for the densities of such arrangements, and furthermore,  solve the problems exactly for the special, yet important, class of unconditional convex bodies."}]