  We introduce a framework unifying the mathematical characterisation of
different measures of general quantum resources and allowing for a systematic
way to define a variety of faithful quantifiers for any given convex quantum
resource theory. The approach allows us to describe many commonly used measures
such as matrix norm-based quantifiers, robustness measures, convex roof-based
measures, and witness-based quantifiers together in a common formalism based on
the convex geometry of the underlying sets of resource-free states. We
establish easily verifiable criteria for a measure to possess desirable
properties such as faithfulness and strong monotonicity under relevant free
operations, and show that many quantifiers obtained in this framework indeed
satisfy them for any considered quantum resource. We derive various bounds and
relations between the measures, generalising and providing significantly
simplified proofs of results found in the resource theories of quantum
entanglement and coherence. We also prove that the quantification of resources
in this framework simplifies for pure states, allowing us to obtain more easily
computable forms of the considered measures and show that several of them are
equal on pure states. Further, we investigate the dual formulation of resource
quantifiers, characterising sets of resource witnesses.
  We present an explicit application of the results to the resource theories of
multi-level coherence, entanglement of Schmidt number k, multipartite
entanglement, as well as magic states, providing insight into the
quantification of the resources and introducing new quantifiers, such as a
measure of entanglement of Schmidt number k which generalises the convex
roof-extended negativity, a measure of k-coherence which generalises the L1
norm of coherence, and a hierarchy of norm-based quantifiers of k-partite
entanglement generalising the greatest cross norm.
