Georgia Institute of Technology College of Sciences

A fundamental result in the geometry of numbers states that any lattice free convex set in R^n has integer width bounded by a function of dimension, i.e. the so called Flatness Theorem for Convex Bodies. This result provides the theoretical basis for the polynomial solvability of Integer Programs with a fixed number of (general) integer variables. In this work, we provide a simplified proof of the Flatness Theorem with tighter constants. Our main technical contribution is a new tight bound on the smoothing parameter of a lattice, a concept developed within lattice based cryptography which enables comparisons between certain discrete distributions over integer points with associated continuous Gaussian distributions. Based on joint work with Kai-Min Chung, Feng Hao Liu, and Christopher Peikert.