  The Loomis-Whitney inequality, and the more general Uniform Cover inequality,
bound the volume of a body in terms of a product of the volumes of
lower-dimensional projections of the body. In this paper, we prove stability
versions of these inequalities, showing that when they are close to being
tight, the body in question is close in symmetric difference to a 'box'. Our
results are best possible up to a constant factor depending upon the dimension
alone. Our approach is information theoretic.
  We use our stability result for the Loomis-Whitney inequality to obtain a
stability result for the edge-isoperimetric inequality in the infinite
$d$-dimensional lattice. Namely, we prove that a subset of $\mathbb{Z}^d$ with
small edge-boundary must be close in symmetric difference to a $d$-dimensional
cube. Our bound is, again, best possible up to a constant factor depending upon
$d$ alone.
