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

Norms extremal with respect to the Mahler measure