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?
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