Holmes–Thompson volume

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.[1]

Definition

The Holmes–Thompson 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 ).

Symplectic (coordinate-free) definition

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 (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 .

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 (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.

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 (possibly asymmetric) norm for each point . The Holmes–Thompson volume of a subset AM 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 Holmes–Thompson volume of A 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.[9]

Normalization and comparison with Euclidean and Hausdorff measure

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 . 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

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.). A sampler of Riemann-Finsler geometry. MSRI Publications. 50. Cambridge University Press. pp. 1–48. ISBN 0-521-83181-4. MR 2132656.

  1. Holmes, Raymond D.; Thompson, Anthony Charles (1979). "N-dimensional area and content in Minkowski spaces". Pacific J. Math. 85 (1): 77–110. doi:10.2140/pjm.1979.85.77. MR 0571628.
  2. Sabourau, Stéphane (2010). "Local extremality of the Calabi–Croke sphere for the length of the shortest closed geodesic". Journal of the London Mathematical Society. 82 (3): 549–562. arXiv:0907.2223. doi:10.1112/jlms/jdq045. S2CID 1156703.
  3. Álvarez Paiva, Juan-Carlos; Balacheff, Florent; Tzanev, Kroum (2016). "Isosystolic inequalities for optical hypersurfaces". Advances in Mathematics. 301: 934–972. arXiv:1308.5522. doi:10.1016/j.aim.2016.07.003. S2CID 119175687.
  4. Ivanov, Sergei V. (2010). "Volume Comparison via Boundary Distances". Proceedings of ICM. arXiv:1004.2505.
  5. Ivanov, Sergei V. (2001). "On two-dimensional minimal fillings". Algebra i Analiz (in Russian). 13 (1): 26–38.
  6. Ivanov, Sergei V. (2002). "On two-dimensional minimal fillings". St. Petersburg Math. J. 13 (1): 17–25. MR 1819361.
  7. Ivanov, Sergei V. (2011). "Filling minimality of Finslerian 2-discs". Proc. Steklov Inst. Math. 273 (1): 176–190. arXiv:0910.2257. doi:10.1134/S0081543811040079. S2CID 115167646.
  8. Ivanov, Sergei V. (2013). "Local monotonicity of Riemannian and Finsler volume with respect to boundary distances". Geometriae Dedicata. 164 (2013): 83–96. arXiv:1109.4091. doi:10.1007/s10711-012-9760-y. S2CID 119130237.
  9. "Santaló formula". Encyclopedia of Mathematics.
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