Groupe d’études et de recherche en analyse des décisions

Comparing points in multi-dimensional spaces, even when they are random

Ricardas Zitikis

In many areas of human activity, researchers and practitioners have been comparing Ð rigorously or just intuitively Ð multi-dimensional and even infinite-dimensional objects. As an example, we can think about a teacher comparing the performance of several classes, even when the classes have more than one student. Another example would be a geographer comparing land in different parts of the world, even when samples have more than one observation. Or we can think about the Antoine de Saint-ExupŽryÕs Little Prince attempting to decide on which of the many planets he would like to live. So how do we compare multi-dimensional objects, or points, apart from first mapping them into the real line by calculating, say, the means of their coordinates? Economists, Geographers, Naturalists and Social Scientists have long used Lorenz curves and Gini indices for answering such questions. Mathematicians have many times appealed in their research to such notions as monotone and convex rearrangements of vectors and functions. Interestingly, the Lorenz curves and convex rearrangements appear to be just two different names of same object. With a number of examples and recent results, we shall give an overview of this fascinating research area.