Lattice point inequalities for centered convex bodies

TL;DR

This paper investigates upper bounds on lattice points within convex bodies centered at the origin, providing optimal results for simplices and planar cases, and extending bounds to general convex bodies.

Contribution

It introduces new upper bounds for lattice points in convex bodies with centroid at the origin, highlighting the restrictive nature of the centroid assumption.

Findings

Optimal bounds for simplices and planar convex bodies.

Extended bounds for general convex bodies.

Centroid assumption is more restrictive than interior lattice point count.

Abstract

We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide an upper bound, which extends the centrally symmetric case and which, in particular, shows that the centroid assumption is indeed much more restrictive than an assumption on the number of interior lattice points even for the class of lattice polytopes.