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]

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 ${\displaystyle \|-\|^{*}}$).

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 ${\displaystyle \mathbb {R} ^{2n}}$.

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.

If ${\displaystyle M}$ is a region in coordinate space ${\displaystyle \mathbb {R} ^{n}}$, then the tangent and cotangent spaces at each point ${\displaystyle x\in M}$ can both be identified with ${\displaystyle \mathbb {R} ^{n}}$. The Finsler metric is a continuous function ${\displaystyle F:TM=M\times \mathbb {R} ^{n}\to [0,+\infty )}$ that yields a (possibly asymmetric) norm ${\displaystyle F_{x}:v\in \mathbb {R} ^{n}\mapsto \|v\|_{x}=F(x,v)}$ for each point ${\displaystyle x\in M}$. The Holmes–Thompson volume of a subset A ⊆ M can be computed as

${\displaystyle \operatorname {Vol} _{\textrm {HT}}(A)=|B^{*}A|=\int _{A}|B_{x}^{*}|\,\mathrm {d} \operatorname {Vol_{n}} (x)}$

where for each point ${\displaystyle x\in M}$, the set ${\displaystyle B_{x}^{*}\subseteq \mathbb {R} ^{n}}$ is the dual unit ball of ${\displaystyle F_{x}}$ (the unit ball of the dual norm ${\displaystyle F_{x}^{*}=\|-\|_{x}^{*}}$), the bars ${\displaystyle |-|}$ denote the usual volume of a subset in coordinate space, and ${\displaystyle \mathrm {d} \operatorname {Vol_{n}} (x)}$ is the product of all n coordinate differentials ${\displaystyle \mathrm {d} x_{i}}$.

This formula follows, again, from the fact that the 2n-form ${\displaystyle \textstyle {{\frac {1}{n!}}\omega ^{n}}}$ is equal (up to a sign) to the product of the differentials of all ${\displaystyle n}$ coordinates ${\displaystyle \mathrm {x} _{i}}$ and their dual coordinates ${\displaystyle \xi _{i}}$. The Holmes–Thompson volume of A is then equal to the usual volume of the subset ${\displaystyle B^{*}A=\{(x,\xi )\in M\times \mathbb {R} ^{n}:\xi \in B_{x}^{*}\}}$ of ${\displaystyle \mathbb {R} ^{2n}}$.

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]

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).

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.