In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces. It was introduced by Raymond D. Holmes and Anthony Charles Thompson.
The Holmes–Thompson volume can be defined without coordinates: if is a measurable set in an n-dimensional real normed space then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form over the set, where is the standard symplectic form on the vector space and is the dual unit ball of. This definition is consistent with the previous one, because if each vector is given linear coordinates and each covector is given the dual coordinates , then the standard symplectic form is, and the volume form is whose integral over the set is just the usual volume of the set in the coordinate space.
Volume in Finsler manifolds
More generally, the Holmes–Thompson volume of a measurable set in a Finsler manifold can be defined as where and is the standard symplectic form on the cotangent bundle. Holmes–Thompson's definition of volume is appropriate for establishing links between the total volume of a manifold and the length of the geodesics contained in it because, according to Liouville's theorem, the geodesic flow preserves the symplectic volume of sets in the cotangent bundle.
Computation using coordinates
If is a region in coordinate space, then the tangent and cotangent spaces at each point can both be identified with. The Finsler metric is a continuous function that yields a norm for each point. The Holmes–Thompson volume of a subset can be computed as where for each point, the set is the dual unit ball of , the bars denote the usual volume of a subset in coordinate space, and is the product of all coordinate differentials. This formula follows, again, from the fact that the -form is equal to the product of the differentials of all coordinates and their dual coordinates. The Holmes–Thompson volume of is then equal to the usual volume of the subset of.
If is a simple region in a Finsler manifold, then its Holmes–Thompson volume can be computed in terms of the path-length distance 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.
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 is normalized, then it never exceeds the Hausdorff measure. This is a consequence of the Blaschke-Santaló inequality. The equality holds if and only ifthe space is Euclidean.
