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


  • Andrzej Tomski University of Silesia, Institute of Mathematics, Bankowa 14, 40-007 Katowice, Poland
  • Szymon Łukaszyk Łukaszyk Patent Attorneys, Głowackiego 8, 40-052 Katowice, Poland



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


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.

Author Biographies

Andrzej Tomski, University of Silesia, Institute of Mathematics, Bankowa 14, 40-007 Katowice, Poland

ORCID 0000-0001-6944-0600

Szymon Łukaszyk, Łukaszyk Patent Attorneys, Głowackiego 8, 40-052 Katowice, Poland

ORCID: 0000-0001-6644-2467


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2024-05-18 — Updated on 2024-06-08


How to Cite

Tomski, A., & Łukaszyk, S. (2024). Reply to "Various issues around the L1-norm distance". IPI Letters, 2(2), 1–8. (Original work published May 18, 2024)