  We propose a new framework that unifies different fairness measures into a
general, parameterized class of convex fairness measures suitable for
optimization contexts. First, we propose a new class of order-based fairness
measures, discuss their properties, and derive an axiomatic characterization
for such measures. Then, we introduce the class of convex fairness measures,
discuss their properties, and derive an equivalent dual representation of these
measures as a robustified order-based fairness measure over their dual sets.
Importantly, this dual representation renders a unified mathematical expression
and an alternative geometric characterization for convex fairness measures
through their dual sets. Moreover, it allows us to develop a unified framework
for optimization problems with a convex fairness measure objective or
constraint, including unified reformulations and solution methods. In addition,
we provide stability results that quantify the impact of employing different
convex fairness measures on the optimal value and solution of the resulting
fairness-promoting optimization problem. Finally, we present numerical results
demonstrating the computational efficiency of our unified framework over
traditional ones and illustrating our stability results.
