Geodesic
Barbilian's metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1059-1068

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In the present paper we answer two questions raised by Barbilian in 1960. First, we study how far can the hypothesis of Barbilian's metrization procedure can be relaxed. Then, we prove that Barbilian's metrization procedure in the plane generates either Riemannian metrics or Lagrance generalized metrics not reducible to Finslerian or Langrangian metrics.
Classification : 30C62, 51K05, 53B40, 53C60
Keywords: {Riemannian metrics, Finslerian metrics, Lagrangian metrics, Lagrange generalized metrics, Barbilian's metrization procedure, Apollonian metric} 
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     author = {Boskoff, Wladimir G. and Suceav\u{a}, Bogdan D.},
     title = {Barbilian's metrization procedure in the plane yields either {Riemannian} or {Lagrange} generalized metrics},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1059--1068},
     publisher = {mathdoc},
     volume = {58},
     number = {4},
     year = {2008},
     mrnumber = {2471165},
     zbl = {1174.53015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008__58_4_a12/}
}
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Boskoff, Wladimir G.; Suceavă, Bogdan D. Barbilian's metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1059-1068. http://geodesic.mathdoc.fr/item/CMJ_2008__58_4_a12/