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We develop a technique that can be applied to provide improved upper boundsfor two important questions in linear integer optimization.  - Proximity bounds: Given an optimal vertex solution for the linearrelaxation, how far away is the nearest optimal integer solution (if oneexists)?  - Flatness bounds: If a polyhedron contains no integer point, what is thesmallest 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 prooftechnique that has been recently introduced by the authors. A key technicallemma underlying our technique concerns the areas of certain convex polygons inthe plane: if a polygon $K\subseteq\mathbb{R}^2$ satisfies $\tau K \subseteqK^{\circ}$, where $\tau$ denotes $90^{\circ}$ counterclockwise rotation and$K^{\circ}$ denotes the polar of $K$, then the area of $K^{\circ}$ is at least3.

Proximity and flatness bounds for linear integer optimization