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In this paper we would like to give an instance of how (real!) Finsler geometry may find applications in problems of complex analysis. In particular we shall show how the understanding of the classification of Finsler manifolds with vanishing horizontal flag curvature together with the theory of complex Monge-Ampère equation enables us to formulate characterizations, up to biholomorphic maps, of special classes of domains in C. Let us start with a simple example. Let D ⊂ R be a smooth strictly convex domain and suppose that 0 ∈ D is the baricenter of D. The Minkowski functional μ of D relative to the baricenter 0 is defined by μ(0) = 0 and for y ∈ R \ {0} by:

Finsler metrics of constant curvature and the characterization of tube domains