According to the Wikipedia article about Income inequality metrics, Gini index have the next disadvantage:
As a disadvantage, the Gini index only maps a number to the properties of a diagram, but the diagram itself is not based on any model of a distribution process. The "meaning" of the Gini index only can be understood empirically. Additionally the Gini does not capture where in the distribution the inequality occurs. As a result, two very different distributions of income can have the same Gini index.
So, I want to ask - what do all the distributions that have the same Gini index have in common? (Empirically, they produce the same area under the Lorentz curve, but what does that mean?)