In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces (of the same dimension).
It was introduced by Raymond D. Holmes and Anthony Charles Thompson.
is defined as the 2n-dimensional measure of the product set
(the unit ball of the dual norm
is a measurable set in an n-dimensional real normed space
is the standard symplectic form on the vector space
This definition is consistent with the previous one, because if each point
, and the volume form is whose integral over the set
is just the usual volume of the set in the coordinate space
More generally, the Holmes–Thompson volume of a measurable set
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 (shortest curves) contained in it (such as systolic inequalities[2][3] and filling volumes[4][5][6][7][8]) because, according to Liouville's theorem, the geodesic flow preserves the symplectic volume of sets in the cotangent bundle.
, then the tangent and cotangent spaces at each point
The Finsler metric is a continuous function
that yields a (possibly asymmetric) norm
The Holmes–Thompson volume of a subset A ⊆ M can be computed as where for each point
is the dual unit ball of
(the unit ball of the dual norm
denote the usual volume of a subset in coordinate space, and
is equal (up to a sign) to the product of the differentials of all
is a simple region in a Finsler manifold (that is, a region homeomorphic to a ball, with convex boundary and a unique geodesic along
), then its Holmes–Thompson volume can be computed in terms of the path-length distance (along
using Santaló's formula, which in turn is based on the fact that the geodesic flow on the cotangent bundle is Hamiltonian.
[9] The original authors used[1] 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
If the Holmes–Thompson volume in normed spaces (or Finsler manifolds) is normalized, then it never exceeds the Hausdorff measure.
The equality holds if and only if the space is Euclidean (or a Riemannian manifold).
Álvarez-Paiva, Juan-Carlos; Thompson, Anthony C. (2004).
"Chapter 1: Volumes on Normed and Finsler Spaces" (PDF).
In Bao, David; Bryant, Robert L.; Chern, Shiing-Shen; Shen, Zhongmin (eds.).