Local monotonicity of Riemannian and Finsler volume with respect to boundary distances

Abstract

We show that the volume of a simple Riemannian metric on D n is locally monotone with respect to its boundary distance function. Namely if g is a simple metric on D n and g′ is sufficiently close to g and induces boundary distances greater or equal to those of g, then vol(D n, g′) ≥ vol(D n, g). Furthermore, the same holds for Finsler metrics and the Holmes–Thompson definition of volume. As an application, we give a new proof of injectivity of the geodesic ray transform for a simple Finsler metric.