Various issues around the L1-norm distance

TL;DR

This paper explores the properties, interpretations, and applications of the L1-norm distance between real-valued random variables, providing new formulas, theoretical insights, and connections to economics and physics.

Contribution

It offers new formulas for the L1-norm distance under independence for Gaussian and uniform variables, and introduces a rigorous, accessible overview of the optimal transport problem with L1 cost.

Findings

Triangle inequality holds for normalized E|X-Y| under independence.

Derived explicit formulas for E|X-Y| for Gaussian and uniform distributions.

Provided an accessible introduction to the optimal transport problem with L1 cost.

Abstract

Beyond the new results mentioned hereafter, this article aims at familiarizing researchers working in applied fields -- such as physics or economics -- with notions or formulas that they use daily without always identifying all their theoretical features or potentialities. Various situations where the L1-norm distance E|X-Y| between real-valued random variables intervene are closely examined. The axiomatic surrounding this distance is also explored. We constantly try to build bridges between the concrete uses of E|X-Y| and the underlying probabilistic model. An alternative interpretation of this distance is also examined, as well as its relation to the Gini index (economics) and the Lukaszyk-Karmovsky distance (physics). The main contributions are the following: (a) We show that under independence, triangle inequality holds for the normalized form E|X-Y|/(E|X| + E|Y|). (b) In order to present a concrete advance, we determine the analytic form of E|X-Y| and of its normalized expression when X and Y are independent with Gaussian or uniform distribution. The resulting formulas generalize relevant tools already in use in areas such as physics and economics. (c) We propose with all the required rigor a brief one-dimensional introduction to the optimal transport problem, essentially for a L1 cost function. The chosen illustrations and examples should be of great help for newcomers to the field. New proofs and new results are proposed.