Lattice Size and Generalized Basis Reduction in Dimension 3

TL;DR

This paper extends lattice size computation and basis reduction algorithms from two to three dimensions, providing faster methods to analyze convex bodies and their lattice properties.

Contribution

It introduces a new fast algorithm for lattice reduction in three dimensions based on generalized basis reduction methods.

Findings

The generalized Gauss algorithm computes lattice size in 2D efficiently.

The paper extends lattice reduction techniques to 3D for convex bodies.

It provides a method to recover successive minima and lattice size from reduced bases.

Abstract

The lattice size of a lattice polytope $P$ was defined and studied by Schicho, and Castryck and Cools. They provided an "onion skins" algorithm for computing the lattice size of a lattice polygon $P$ in $\mathbb{R}^2$ based on passing successively to the convex hull of the interior lattice points of $P$. We explain the connection of the lattice size to the successive minima of $K=\left(P+(-P)\right)^\ast$ and to the lattice reduction with respect to the general norm that corresponds to $K$. It follows that the generalized Gauss algorithm of Kaib and Schnorr (which is faster than the "onion skins" algorithm) computes the lattice size of any convex body in $\mathbb{R}^2$. We extend the work of Kaib and Schnorr to dimension 3, providing a fast algorithm for lattice reduction with respect to the general norm defined by a convex origin-symmetric body $K\subset\mathbb{R}^3$. We also explain how to recover the successive minima of $K$ and the lattice size of $P$ from the obtained reduced basis and therefore provide a fast algorithm for computing the lattice size of any convex body $P\subset\mathbb{R}^3$.