SOME REMARKS ON THE POMPEIU-HAUSDORFF DISTANCE BETWEEN ORDER INTERVALS

摘要

The distance between the subsets of R n was introduced by the Romanian mathe- matician Dimitrie Pompeiu in 1905 in his Ph. D. Thesis [10] (see also [3]). Is worth noting that the concept of metric space appeared in the Ph. D. thesis of Maurice Fréchet [6] published a little bit later in 1906. Felix Hausdorff studied the notion of set distance in the setting of metric spaces in his book from 1914 [7]. Hausdorff quoted Pompeiu as the author of the notion of distance between sets, but not in the main text of the book, only in the final notes, as was the procedure at that time to indicate the references ([8], p. 343). And thus the distance between sets came to be called the Hausdorff distance. We will call this distance Pompeiu-Hausdorff distance like in the book of R. Tyrrell Rockafellar and Roger J.-B. Wets “Variational Analysis” ([12], p. 144).