Towards Strong Reverse Minkowskitype Inequalities for Lattices
Abstract
We present a natural reverse Minkowskitype inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits a surprising wealth of connections to various areas in mathematics and computer science, including a conjecture motivated by integer programming by Kannan and Lovász (Annals of Math. 1988), a question from additive combinatorics asked by Green, a question on Brownian motions asked by SaloffCoste (Colloq. Math. 2010), a theorem by Milman and Pisier from convex geometry (Ann. Probab. 1987), worstcase to averagecase reductions in latticebased cryptography, and more. We present these connections, provide evidence for the conjecture, and discuss possible approaches towards a proof. Our main technical contribution is in proving that our conjecture implies the $\ell_2$ case of the Kannan and Lovász conjecture. The proof relies on a novel convex relaxation for the covering radius, and a rounding procedure for based on "uncrossing" lattice subspaces.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.06913
 Bibcode:
 2016arXiv160606913D
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Computational Complexity;
 Mathematics  Functional Analysis;
 Mathematics  Probability