  We study the mathematical and economic structure of the Kolkata (k) index of
income inequality. We show that the k-index always exists and is a unique fixed
point of the complementary Lorenz function, where the Lorenz function itself
gives the fraction of cumulative income possessed by the cumulative fraction of
population (when arranged from poorer to richer). We show that the k-index
generalizes Pareto's 80/20 rule. Although the k and Pietra indices both split
the society into two groups, we show that k-index is a more intensive measure
for the poor-rich split. We compare the normalized k-index with the Gini
coefficient and the Pietra index and discuss when they coincide. We establish
that for any income distribution the value of Gini coefficient is no less than
that of the Pietra index and the value of the Pietra index is no less than that
of the normalized k-index. While the Gini coefficient and the Pietra index are
affected by transfers exclusively among the rich or among the poor, the k-index
is only affected by transfers across the two groups.
