98 research outputs found
A remark on perimeter-diameter and perimeter-circumradius inequalities under lattice constraints
In this note, we study several inequalities involving geometric functionals
for lattice point-free planar convex sets. We focus on the previously not
addressed cases perimeter--diameter and perimeter--circumradius
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
Notes on lattice points of zonotopes and lattice-face polytopes
Minkowski's second theorem on successive minima gives an upper bound on the
volume of a convex body in terms of its successive minima. We study the problem
to generalize Minkowski's bound by replacing the volume by the lattice point
enumerator of a convex body. In this context we are interested in bounds on the
coefficients of Ehrhart polynomials of lattice polytopes via the successive
minima. Our results for lattice zonotopes and lattice-face polytopes imply, in
particular, that for 0-symmetric lattice-face polytopes and lattice
parallelepipeds the volume can be replaced by the lattice point enumerator.Comment: 16 pages, incorporated referee remarks, corrected proof of Theorem
1.2, added new co-autho
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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 Mahlerconjecture.
Based on the covering minima of Kannan & Lovász and a problem
posed by Fejes Tóth, 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
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