  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.
