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 $.

Has there been any work done to find short vectors based on other, potentially higher, norms? Is this a meaningful question?