Proximity and flatness bounds for linear integer optimization

We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. - Proximity bounds: Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists)? - Flatness bounds: If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, non-zero, normal vector that intersect the polyhedron? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane: if a polygon $K\subseteq {\mathbb{R}}^2$ satisfies $\tau K\subseteq K^{\circ }$, where $\tau$ denotes ${90}^{\circ }$ counterclockwise rotation and $K^{\circ }$ denotes the polar of $K$, then the area of $K^{\circ }$ is at least 3.