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 ${\displaystyle \operatorname {Vol} _{\text{HT}}(A)}$ of a measurable set ${\displaystyle A\subseteq R^{n}}$ in a normed space ${\displaystyle (\mathbb {R} ^{n},\|-\|)}$ is defined as the 2n-dimensional measure of the product set ${\displaystyle A\times B^{*},}$ where ${\displaystyle B^{*}\subseteq \mathbb {R} ^{n}}$ is the dual unit ball of ${\displaystyle \|-\|}$ (the unit ball of the dual norm ${\displaystyle \|-\|^{*}}$).

Symplectic (coordinate-free) definition

The Holmes–Thompson volume can be defined without coordinates: if ${\displaystyle A\subseteq V}$ is a measurable set in an n-dimensional real normed space ${\displaystyle (V,\|-\|),}$ then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form ${\displaystyle {\frac {1}{n!}}\overbrace {\omega \wedge \cdots \wedge \omega } ^{n}}$ over the set ${\displaystyle A\times B^{*}}$,

${\displaystyle \operatorname {Vol} _{HT}(A)=\left|\int _{A\times B^{*}}{\frac {1}{n!}}\omega ^{n}\right|}$

where ${\displaystyle \omega }$ is the standard symplectic form on the vector space ${\displaystyle V\times V^{*}}$ and ${\displaystyle B^{*}\subseteq V^{*}}$ is the dual unit ball of ${\displaystyle \|-\|}$.

This definition is consistent with the previous one, because if each point ${\displaystyle x\in V}$ is given linear coordinates ${\displaystyle (x_{i})_{0\leq i<n}}$ and each covector ${\displaystyle \xi \in V^{*}}$ is given the dual coordinates ${\displaystyle (xi_{i})_{0\leq i<n}}$ (so that ${\displaystyle \xi (x)=\sum _{i}\xi _{i}x_{i}}$), then the standard symplectic form is ${\displaystyle \omega =\sum _{i}\mathrm {d} x_{i}\wedge \mathrm {d} \xi _{i}}$, and the volume form is

${\displaystyle {\frac {1}{n!}}\omega ^{n}=\pm \;\mathrm {d} x_{0}\wedge \dots \wedge \mathrm {d} x_{n-1}\wedge \mathrm {d} \xi _{0}\wedge \dots \wedge \mathrm {d} \xi _{n-1},}$

whose integral over the set ${\displaystyle A\times B^{*}\subseteq V\times V^{*}\cong \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$ is just the usual volume of the set in the coordinate space ${\displaystyle \mathbb {R} ^{2n}}$.

Volume in Finsler manifolds

More generally, the Holmes–Thompson volume of a measurable set ${\displaystyle A}$ in a Finsler manifold ${\displaystyle (M,F)}$ can be defined as

${\displaystyle \operatorname {Vol} _{\text{HT}}(A):=\int _{B^{*}A}{\frac {1}{n!}}\omega ^{n},}$

where ${\displaystyle B^{*}A=\{(x,p)\in \mathrm {T} ^{*}M:\ x\in A{\text{ and }}\xi \in \mathrm {T} _{x}^{*}M{\text{ with }}\|\xi \|_{x}^{*}\leq 1\}}$ and ${\displaystyle \omega }$ is the standard symplectic form on the cotangent bundle ${\displaystyle \mathrm {T} ^{*}M}$. 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.

Santaló's formula

If ${\displaystyle A}$ is a simple region in a Finsler manifold (that is, a region homeomorphic to a ball, with convex boundary and a unique geodesic along ${\displaystyle A}$ joining each pair of points of ${\displaystyle A}$), then its Holmes–Thompson volume can be computed in terms of the path-length distance (along ${\displaystyle A}$) between the boundary points of ${\displaystyle A}$ 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 ${\displaystyle (\mathbb {R} ^{n},\|-\|_{2})}$. 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. Vol. 50. Cambridge University Press. pp. 1–48. ISBN 0-521-83181-4. MR 2132656.