  The main subject of this expository paper is a connection between Gromov's
filling volumes and a boundary rigidity problem of determining a Riemannian
metric in a compact domain by its boundary distance function. A fruitful
approach is to represent Riemannian metrics by minimal surfaces in a Banach
space and to prove rigidity by studying the equality case in a filling volume
inequality. I discuss recent results obtained with this approach and related
problems in Finsler geometry.
