One Hundred Years Since the Introduction of the Set Distance by Dimitrie Pompeiu

This paper recalls the work of D. Pompeiu who introduced the notion of set distance in his thesis published one century ago. The notion was further studied by F. Hausdorff, C. Kuratowski who acknowledged in their books the contribution of Pompeiu and it is frequently called the Hausdorff distance. keywords: Hausdorff distance, Hausdorff-Pompeiu distance, Pompeiu functions, Pompeiu conjecture, Schiffer conjecture 1 The thesis of Pompeiu On March 31-st, 1905, Dimitrie Pompeiu (1873-1954), a distinguished Romanian mathematician, defended at the Faculté de Sciences de Paris his Ph.D. work Sur la continuité des fonctions de variables complexes . The comission was chaired by H. Poincaré and included G. Koenigs and E. Goursat and the work was published the same year in [14]. P. Montel, a wellknown French mathematician, has appreciated the remarkable contribution of Pompeiu’s thesis by the words: Pour un coup d’essai, c’est un coup de mâıtre, [1]. Ideas, notions and results discussed in the thesis of Pompeiu have attracted the interest of important contemporary mathematicians: ∗ [email protected] “Gh. Asachi” University, Department of Mathematics, Iasi, Romania † [email protected] Institute of Mathematics, Romanian Academy, Bucharest, Romania