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A new metric on the open 2-dimensional unit disk is defined making it ageodesically complete metric space whose geodesic lines are precisely theEuclidean straight lines. Moreover, it is shown that the unit disk with thisnew metric is not isometric to any hyperbolic model of constant negativecurvature, nor to any convex domain in R2 equipped with its Hilbert metric.

A specific model of Hilbert geometry on the unit disc