  We address the problem of finding the optimal common resource for an
arbitrary family of target states in quantum resource theories based on
majorization, that is, theories whose conversion law between resources is
determined by a majorization relationship, such as it happens with
entanglement, coherence or purity. We provide a conclusive answer to this
problem by appealing to the completeness property of the majorization lattice.
We give a proof of this property that relies heavily on the more geometric
construction provided by the Lorenz curves, which allows to explicitly obtain
the corresponding infimum and supremum. Our framework includes the case of
possibly non-denumerable sets of target states (i.e. targets sets described by
continuous parameters). In addition, we show that a notion of approximate
majorization, which has recently found application in quantum thermodynamics,
is in close relation with the completeness of this lattice. Finally, we provide
some examples of optimal common resources within the resource theory of quantum
coherence.
