Bounds on the Lattice Point Enumerator via Slices and Projections

Abstract

Gardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

On Rogers-Shephard type inequalities for the lattice point enumerator

“…Apart from the above-mentioned discrete analogues of the Brunn-Minkowski inequality, various discrete counterparts, for the lattice point enumerator G n (•), of results in Convex Geometry were recently proven. Some examples of such results are Koldobsky's slicing inequality [2], Meyer's inequality [19] and an isoperimetric type inequality [24]. We refer the reader to these articles and the references therein for other connected problems, questions and results.…”

Inequalities for sections and projections of convex bodies

“…Another very interesting bound on the cardinality of lattice points in sections of convex bodies is inspired by Meyer's inequality (3.2). It was proposed by Gardner, Gronchi and Zong [45] and proved by Freyer and Henk in [40]: For any originsymmetric convex body K ⊂ R n there exists a basis b 1 , . .…”

“…, t n ∈ Z n such that (#K) for any origin-symmetric convex body K in R n . Actually, Freyer and Henk [40] removed the condition of K being symmetric in the statement above (the condition cannot be removed in the discrete analogue of Meyer's inequality). Moreover, they were able to prove that in the symmetric case the O(n 2 )-term can be replaced by O(n).…”

Polynomial Bounds in Koldobsky's Discrete Slicing Problem

“…The main reason for this exponential gap is the unfortunate circumstance that, even though K is origin-symmetric, the maximal (with respect to lattice points) hyperplane section does not need to pass through the origin. In fact, given a direction y = 0 in R n the maximal affine hyperplane section of K orthogonal to y might contain 2 n times as many lattice points as the parallel section through the origin (see, e.g., [11,Lemma 1.3]).…”

“…On the other hand it is known [11,Theorem 1.4] that for K ∈ K n os there always exists an affine hyperplane A such that…”