Volumes and areas of Lipschitz metrics

Methods of estimating (Riemannian and Finsler) filling volumes by using nonexpanding maps to Banach spaces of $L^\infty$ type are developed and generalized. For every Finsler volume functional (such as the Busemann volume or the Holmes–Thompson volume), a natural extension is constructed from the class of Finsler metrics to all Lipschitz metrics, and the notion of area is defined for Lipschitz surfaces in a Banach space. A correspondence is established between minimal fillings and minimal surfaces in $L^\infty$ type spaces. A Finsler volume functional for which the Riemannian and the Finsler filling volumes are equal is introduced; it is proved that this functional is semielliptic.