LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $\varepsilon$ as its shortest vector, $\varepsilon \in \mathbb{Z}^n$, find a short vector s.t. $b \in \mathbb{Z}^n, ||b||_2 < ||c^n \varepsilon||_2$.