  We introduce and study a generalization of majorization called relative
submajorization and show that it has many applications to the resource theories
of thermodynamics, bipartite entanglement, and quantum coherence. In
particular, we show that relative submajorization characterizes both the
probability and approximation error that can be obtained when transforming one
resource to another, also when assisted by additional standard resources such
as useful work or maximally-entangled states. These characterizations have a
geometric formulation as the ratios or differences, respectively, between the
Lorenz curves associated with the input and output resources. We also find
several interesting bounds on the reversibility of a given transformation in
terms of the properties of the forward transformation. The main technical tool
used to establish these results is linear programming duality.
