1,525 research outputs found
On densities of lattice arrangements intersecting every i-dimensional affine subspace
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.Comment: 19 page
Restricted Successive Minima
We give bounds on the successive minima of an -symmetric convex body under
the restriction that the lattice points realizing the successive minima are not
contained in a collection of forbidden sublattices. Our investigations extend
former results to forbidden full-dimensional lattices, to all successive minima
and complement former results in the lower dimensional case.Comment: 11 pages, Abstract and Introduction revised in view of new added
reference
Mahler's work on the geometry of numbers
Mahler has written many papers on the geometry of numbers. Arguably, his most
influential achievements in this area are his compactness theorem for lattices,
his work on star bodies and their critical lattices, and his estimates for the
successive minima of reciprocal convex bodies and compound convex bodies. We
give a, by far not complete, overview of Mahler's work on these topics and
their impact.Comment: 17 pages. This paper will appear in "Mahler Selecta", a volume
dedicated to the work of Kurt Mahler and its impac
Integer Knapsacks: Average Behavior of the Frobenius Numbers
The main result of the paper shows that the asymptotic growth of the
Frobenius number in average is significantly slower than the growth of the
maximum Frobenius number
Densest Lattice Packings of 3-Polytopes
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies
we present an efficient algorithm for computing the density of a densest
lattice packing of an arbitrary 3-polytope. As an application we calculate
densest lattice packings of all regular and Archimedean polytopes.Comment: 37 page
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