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I measure the weighted distance between two vector $a$ and $b$ with dimensions weighted by vector $w$ in $\mathbb{R}^n$, by the weighted 1-norm: $$N_1 = \sum_i^n |w_i(a-b)_i| = \sum_i^n |w_i||(a-b)_i|$$

A relaxation (by relaxing $w_i$) of this norm, that works well for me as input for my classifier, is: $$N_{1,R} = \sum_i^n w_i|(a-b)_i|$$

Could someone point me to a reference of this $N_{1,R}$?

Thank you very much for your help and suggestions.

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