dual coordinates . The Holmes–Thompson volume of is then equal to the usual volume of the subset of . Santaló's formula If is a simple region in a Finsler manifold (that is, a region homeomorphic to a ball, with convex boundary and a unique geodesic along joining each pair of points of ), then its Holmes–Thompson volume can be computed in terms of the path-length distance (along ) between the boundary points of using Santaló's formula, which in turn is based on the fact that the geodesic flow on the cotangent bundle is Hamiltonian. Normalization and comparison with Euclidean and Hausdorff measure The original authors used a different normalization for Holmes–Thompson volume. They divided the value given here by the volume of the Euclidean n-ball, to make Holmes–Thompson volume coincide with the product measure in the standard Euclidean space . This article does not follow that convention. If the Holmes–Thompson volume in normed spaces (or Finsler manifolds) is normalized, then it never exceeds the Hausdorff measure. This is a consequence of the Blaschke-Santaló inequality. The equality holds if and only if the space is Euclidean (or a Riemannian manifold). References Normed spaces Differential geometry Measure theory Finsler geometry Integral geometry Systolic geometry