77 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
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 diversities and finite dimensional Banach spaces
A diversity in is a function defined over every finite set of
points of mapped onto , with the properties that
if and only if and , for every finite sets with . Its importance
relies in the fact that, amongst others, they generalize the notion of metric
distance.
We characterize when a diversity defined over , , is
Banach-embeddable, i.e. when there exist points , , and a
symmetric, convex, and compact set such that
, where
denotes the circumradius of with respect to . Moreover, we also
characterize when a diversity is a Banach diversity, i.e. when
, for every finite set , where is an
-dimensional, symmetric, convex, and compact set
Narrowing the gaps of the missing Blaschke-Santal\'o diagrams
We solve several new sharp inequalities relating three quantities amongst the
area, perimeter, inradius, circumradius, diameter, and minimal width of planar
convex bodies. As a consequence, we narrow the missing gaps in each of the
missing planar Blaschke-Santal\'o diagrams. Furthermore, we extend some of
those sharp inequalities into higher dimensions, by replacing either the
perimeter by the mean width or the area by the volume
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