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?



Other norms for lattice reduction techniques (LLL, PSLQ)?
The state of the art (of the possible) is covered in Khot's paper ["Inapproximability Results for Computational Problems on Lattices"](https://doi.org/10.1007/978-3-642-02295-1_14). [Here is a link](https://books.google.com/books?id=9d75m1L_GIgC&pg=PA456#v=onepage&q&f=false) to a brief section on $\ell^p$ norms.


