$q$-norms are really norms
  Replacing the triangle inequality, in the definition of a norm, by $|x + y| ^{q}\leq 2^{q-1}(|x| ^{q} + |y| ^{q}) $, we introduce the notion of a q-norm. We establish that every q-norm is a norm in the usual sense, and that the converse is true as well. 
H. Belbachir, M. Mirzavaziri and M. S. Moslehian