  In a previous paper, the authors introduced several vector space norms on the
space of algebraic numbers modulo torsion which corresponded to the Mahler
measure on a certain class of numbers and allowed the authors to formulate L^p
Lehmer conjectures which were equivalent to their classical counterparts. In
this paper, we introduce and study several analogous norms which are
constructed in order to satisfy an extremal property with respect to the Mahler
measure. These norms are a natural generalization of the metric Mahler measure
introduced by Dubickas and Smyth. We evaluate these norms on certain classes of
algebraic numbers and prove that the infimum in the construction is achieved in
a certain finite dimensional space.
