# On the Standard Lattices

## TL;DR

This paper investigates the conditions under which lattices are standard across different norms and dimensions, establishing that all Euclidean lattices are standard up to dimension 4, with specific examples and exceptions for other norms.

## Contribution

It proves that all Euclidean lattices are standard in dimensions up to 4 and provides examples of non-standard lattices in higher dimensions for certain norms.

## Key findings

- All Euclidean lattices are standard for n ≤ 4.
- Lattices of dimensions 1 and 2 are standard under any norm.
- Existence of non-standard lattices in higher dimensions with L^1 norm.

## Abstract

A lattice in the Euclidean space is standard if it has a basis consisting vectors whose norms equal to the length in its successive minima. In this paper, it is shown that with the $L^2$ norm all lattices of dimension $n$ are standard if and only if $n\leqslant 4$. It is also proved that with an arbitrary norm, every lattice of dimensions 1 and 2 is standard. An example of non-standard lattice of dimension $n\geqslant 3$ is given when the lattice is with the $L^1$ norm.