# Relaxation of weighted norm distance?

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.