Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Let d and k be integers with 1 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover the intersection of Lambda with K. We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For d > =3 and epsilon in (0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n, m the following statement is true. There is a set of n points in R^d and an arrangement of m hyperplanes in R^d with no K_(r,r) in their incidence graph and with at least Omega((mn)^(1-(2d+3)/((d+2)(d+3)) - epsilon)) incidences if d is odd and Omega((mn)^(1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - epsilon)) incidences if d is even

Minkowski’s successive minima in convex and discrete geometry

In this short survey we want to present some of the impact of Minkowski’s successive minima within Convex and Discrete Geometry. Originally related to the volume of an o-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel’s lemma

Tropical types and associated cellular resolutions

An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the tropical torus induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.Comment: minor correction