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Keywords:
{Riemannian metrics, Finslerian metrics, Lagrangian metrics, Lagrange generalized metrics, Barbilian's metrization procedure, Apollonian metric}
{Riemannian metrics, Finslerian metrics, Lagrangian metrics, Lagrange generalized metrics, Barbilian's metrization procedure, Apollonian metric}
Summary:
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.
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