Geometric stability via information theory

@article{Ellis2015GeometricSV,
  title={Geometric stability via information theory},
  author={David Ellis and Ehud Friedgut and Guy Kindler and Amir Yehudayoff},
  journal={ArXiv},
  year={2015},
  volume={abs/1510.00258}
}
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… 
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