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.
Highlights
One of the central questions in Geometric Tomography is to determine or to reconstruct a set K in the n-dimensional Euclidean space Rn by some of its lower dimensional “structures”
This may be regarded as a lattice version of the well-known slicing problem for volumes, asking for the correct order of a constant c such that for all centered convex bodies K ∈ Kn there exists a u ∈ Rn \ {0} such that(n−1)/n ≤ c vol(K ∩ u⊥)
If K is contained in a k-dimensional space F, we denote by volk K its k-dimensional Lebesgue measure in F. It is a famous open problem in Convex Geometry to find the best possible lower bound on the volume product vol K · vol K, where K ∈ Kons
Summary
One of the central questions in Geometric Tomography is to determine or to reconstruct a set K in the n-dimensional Euclidean space Rn by some of its lower dimensional “structures” (see [13]). This may be regarded as a lattice version of the well-known slicing problem for volumes, asking for the correct order of a constant c such that for all centered convex bodies K ∈ Kn there exists a u ∈ Rn \ {0} such that (vol K )(n−1)/n ≤ c vol(K ∩ u⊥). To this day, the best known bound is of order n1/4 (cf [23]). The proof of Theorem 1.6 is given in Sect. 4, and in the final section we discuss improvements for the special class of unconditional bodies
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