Geometriae Dedicata

, Volume 164, Issue 1, pp 83–96 | Cite as

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

  • Sergei Ivanov
Original Paper


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.


Boundary distance function Minimal filling Finsler metric Holmes–Thompson volume Geodesic ray transform 

Mathematics Subject Classification (1991)

53C60 53C20 44A12 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.St. Petersburg Department, V.A. Steklov Institute of MathematicsRussian Academy of SciencesSaint PetersburgRussia

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