In geometry of normed spaces, the HolmesâThompson 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. [1]
The HolmesâThompson volume of a measurable set in a normed space is defined as the 2n-dimensional measure of the product set where is the dual unit ball of (the unit ball of the dual norm ).
The HolmesâThompson volume can be defined without coordinates: if is a measurable set in an n-dimensional real normed space then its HolmesâThompson 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 point is given linear coordinates and each covector is given the dual coordinates (so that ), 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 .
More generally, the HolmesâThompson volume of a measurable set in a Finsler manifold can be defined as
where and is the standard symplectic form on the cotangent bundle . HolmesâThompson'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.
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 (possibly asymmetric) norm for each point . The HolmesâThompson volume of a subset A â M can be computed as
where for each point , the set is the dual unit ball of (the unit ball of the dual norm ), the bars denote the usual volume of a subset in coordinate space, and is the product of all n coordinate differentials .
This formula follows, again, from the fact that the 2n-form is equal (up to a sign) to the product of the differentials of all coordinates and their dual coordinates . The HolmesâThompson volume of A is then equal to the usual volume of the subset of .
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 HolmesâThompson volume can be computed in terms of the path-length distance (along ) between the boundary points of using SantalĂł'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 HolmesâThompson volume. They divided the value given here by the volume of the Euclidean n-ball, to make HolmesâThompson volume coincide with the product measure in the standard Euclidean space . This article does not follow that convention.
If the HolmesâThompson volume in normed spaces (or Finsler manifolds) is normalized, then it never exceeds the Hausdorff measure. This is a consequence of the Blaschke-SantalĂł inequality. The equality holds if and only if the space is Euclidean (or a Riemannian manifold).
Ălvarez-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.). A sampler of Riemann-Finsler geometry. MSRI Publications. Vol. 50. Cambridge University Press. pp. 1â48. ISBN 0-521-83181-4. MR 2132656.
In geometry of normed spaces, the HolmesâThompson 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. [1]
The HolmesâThompson volume of a measurable set in a normed space is defined as the 2n-dimensional measure of the product set where is the dual unit ball of (the unit ball of the dual norm ).
The HolmesâThompson volume can be defined without coordinates: if is a measurable set in an n-dimensional real normed space then its HolmesâThompson 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 point is given linear coordinates and each covector is given the dual coordinates (so that ), 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 .
More generally, the HolmesâThompson volume of a measurable set in a Finsler manifold can be defined as
where and is the standard symplectic form on the cotangent bundle . HolmesâThompson'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.
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 (possibly asymmetric) norm for each point . The HolmesâThompson volume of a subset A â M can be computed as
where for each point , the set is the dual unit ball of (the unit ball of the dual norm ), the bars denote the usual volume of a subset in coordinate space, and is the product of all n coordinate differentials .
This formula follows, again, from the fact that the 2n-form is equal (up to a sign) to the product of the differentials of all coordinates and their dual coordinates . The HolmesâThompson volume of A is then equal to the usual volume of the subset of .
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 HolmesâThompson volume can be computed in terms of the path-length distance (along ) between the boundary points of using SantalĂł'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 HolmesâThompson volume. They divided the value given here by the volume of the Euclidean n-ball, to make HolmesâThompson volume coincide with the product measure in the standard Euclidean space . This article does not follow that convention.
If the HolmesâThompson volume in normed spaces (or Finsler manifolds) is normalized, then it never exceeds the Hausdorff measure. This is a consequence of the Blaschke-SantalĂł inequality. The equality holds if and only if the space is Euclidean (or a Riemannian manifold).
Ălvarez-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.). A sampler of Riemann-Finsler geometry. MSRI Publications. Vol. 50. Cambridge University Press. pp. 1â48. ISBN 0-521-83181-4. MR 2132656.