  We find the set of extremal points of Lorenz curves with fixed Gini index and
compute the maximal $L^1$-distance between Lorenz curves with given values of
their Gini coefficients. As an application we introduce a bidimensional index
that simultaneously measures relative inequality and dissimilarity between two
populations. This proposal employs the Gini indices of the variables and an
$L^1$-distance between their Lorenz curves. The index takes values in a
right-angled triangle, two of whose sides characterize perfect relative
inequality-expressed by the Lorenz ordering between the underlying
distributions. Further, the hypotenuse represents maximal distance between the
two distributions. As a consequence, we construct a chart to, graphically,
either see the evolution of (relative) inequality and distance between two
income distributions over time or to compare the distribution of income of a
specific population between a fixed time point and a range of years. We prove
the mathematical results behind the above claims and provide a full description
of the asymptotic properties of the plug-in estimator of this index. Finally,
we apply the proposed bidimensional index to several real EU-SILC income
datasets to illustrate its performance in practice.
