# Reply to "Various issues around the L1-norm distance"

## DOI:

https://doi.org/10.59973/ipil.77## Keywords:

Łukaszyk-Karmowski metric, mean absolute difference, identity of indiscernibles## Abstract

A distance function between two random variables or vectors was proposed in 2003 in a Ph.D. dissertation. Initially called a *probability metric*, it is now known as "Łukaszyk-Karmowski metric" or LK-metric and has been successfully applied in various fields of science and technology. It does not satisfy the identity of indiscernible (*Leibniz's law*) axiom of the metric, the ontological axiom also invalidated by the ugly duckling theorem. This note addresses two false claims made in a preprint that LK-metric is the same as the mean absolute difference and that it is ill-defined. The fallacy of the first claim is straightforward: the mean absolute difference is defined solely for independent and identically distributed random variables, contrary to LK-metric. Thus, if one considers *E*|*X*-*X*|, then the random variable *X* must be *independent of itself*, which implies its degenerate probability distribution and *E*|*X*-*X*|=0. If *X* has a degenerate probability distribution, then *Y*, which is identically distributed as *X*, also has a degenerate probability distribution and *E*|*X*-*X*|=0=*E*|*X*-*Y*|, invalidating the second claim.

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