  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.
