Finsler currents

Abstract We propose a slight variant of Ambrosio and Kirchheim’s definition of a metric current. We show that with this new definition it is possible to obtain certain volume functionals from Finsler geometry as mass measures of currents. As an application, we obtain a whole family of extendibly convex $$n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -volume densities. This family includes the circumscribed Riemannian and the mass* volume densities.