Ion Barbu

In 1934, Barbilian published his article[7] describing metrization of a region K, the interior of a simple closed curve J. Let xy denote the Euclidean distance from x to y. Barbilian's function for the distance from a to b in K is

${\displaystyle d(a,b)=\log {\underset {p\in J}{\max }}(pa/pb)+\log {\underset {q\in J}{\max }}(qb/qa).}$

At the University of Missouri in 1938 Leonard Blumenthal wrote Distance Geometry. A Study of the Development of Abstract Metrics,[8] where he used the term "Barbilian spaces" for metric spaces based on Barbilian's function to obtain their metric. And in 1954 the American Mathematical Monthly published an article by Paul J. Kelly on Barbilian's method of metrizing a region bounded by a curve.[9] Barbilian claimed he did not have access to Kelly's publication, but he did read Blumenthal's review of it in Mathematical Reviews and he understood Kelly's construction. This motivated him to write in final form a series of four papers, which appeared after 1958, where the metric geometry of the spaces that today bears his name is investigated thoroughly.

He answered in 1959 with an article[10] which described "a very general procedure of metrization through which the positive functions of two points, on certain sets, can be refined to a distance." Besides Blumenthal and Kelly, articles on "Barbilian spaces" have appeared in the 1990s from Patricia Souza, while Wladimir G. Boskoff, Marian G. Ciucă and Bogdan Suceavă wrote in the 2000s about "Barbilian's metrization procedure".[11] Barbilian indicated in his paper Asupra unui principiu de metrizare that he preferred the term "Apollonian metric space", and articles from Alan F. Beardon, Frederick Gehring and Kari Hag, Peter A. Häströ, Zair Ibragimov and others use that term. According to Suceavă,[12] "Barbilian's metrization procedure is important for at least three reasons: (1) It yields a natural generalization of Poincaré and Beltrami-Klein's hyperbolic geometries; (2) It has been studied in the context of the study of Apollonian metric; (3) Provides a large class of examples of Lagrange generalized metrics irreducible to Riemann, Finsler, or Lagrange metrics."

Barbilian made a contribution to the foundations of geometry with his articles in 1940 and 1941 in Jahresbericht der Deutschen Mathematiker-Vereinigung on projective planes with coordinates from a ring.[13][14] According to Boskoff and Suceavă, this work "inspired research in ring geometries, nowadays associated with his, Hjelmslev's and Klingenberg's names." A more critical stance was taken in 1995 by Ferdinand D. Velkamp:

A systematic study of projective planes over large classes of associative rings was initiated by D. Barbilian. His very general approach in [1940 and 41] remained rather unsatisfactory, however, his axioms were partly of a geometric nature, partly algebraic as pertaining to the ring of coordinates, and there were a number of difficulties which Barbilian could not overcome.[15]

Nevertheless, in 1989 John R. Faulkner wrote an article "Barbilian Planes"[16] that clarified terminology and advanced the study. In his introduction, he wrote:

A classical result from projective geometry is that a Desarguesian projective plane is coordinatized by an associative division ring. A Barbilian plane is a geometric structure which extends the notion of a projective plane and thereby allows a coordinate ring which is not necessarily a division ring. There are advantages ...

• 1956: "Teoria aritmetică a idealelor (în inele necomutative)", Editura Academiei Republicii Populare Romîne, Bucharest. MR0085247
• 1960: "Grupuri cu operatori: Teoremele de descompunere ale algebrei", Editura Academiei Republicii Popular Romîne, Bucharest. MR0125888