Suppose we have a set of vectors of positive real numbers ($a_1,a_2,...,a_m$) and a set of binary ([0,1]) vector($x_1,x_2,...,x_m$). The length of each vector is $n$. We want to optimize the objective function $$ min \sum_i || a_i - a_i \diamond x_i ||_p $$
with some constraints on $x_i$. Here $\diamond$ represent the element wise multiplication operation.
for 1-norm I understand that $\sum_i || a_i - a_i \diamond x_i ||_1 = \sum_i (|| a_i||_1 - ||a_i \diamond x_i ||_1)$. So for 1-norm maximizing $\sum_i || a_i - a_i \diamond x_i ||_1$ corresponds to minimizing $||a_i \diamond x_i ||_1$.
My question is what happen if $p>1$. I think that $|| a_i - a_i \diamond x_i || \geq || a_i|| - ||a_i \diamond x_i ||$ for any norm $p$. If that is true then is maximizing $\sum_i || a_i - a_i \diamond x_i ||_p$ equivalent to maximizing $\sum_i (|| a_i||_p -||a_i \diamond x_i ||_p)$?
Also does it extend to $p=0$ and $p=\infty$?