Exact Flatness Constant for One-Point Convex Bodies and the Discrete Isominwidth Problem: The Planar Case

TL;DR

This paper determines the exact flatness constant for planar convex bodies with at most one interior lattice point, linking it to classical theorems and establishing new inequalities in lattice point enumeration.

Contribution

It computes the exact flatness constant Flt(2, 1) for the planar case and connects it to the discrete isominwidth problem and Minkowski's convex body theorem.

Findings

Flt(2, 1) = 3 for planar convex bodies with at most one interior lattice point

Any such body has lattice width at most three

Establishes an isominwidth inequality related to lattice point enumeration

Abstract

A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it is related to the classical flatness constant as well as a conjectural dual version of Minkowski's convex body theorem due to Makai. Moreover, it is shown that Flt(2, 1) = 3, i.e., any planar convex body with at most one interior point has lattice width at most three. This leads to an isominwidth inequality for the lattice point enumerator of planar convex bodies.