Английская Википедия: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 <math> \operatorname{Vol}_\text{HT}(A) </math> of a measurable set <math> A\subseteq R^n </math> in a normed space <math> (\mathbb{R}^n,\|-\|) </math> is defined as the 2n-dimensional measure of the product set <math> A\times B^*,</math> where <math> B^* \subseteq \mathbb{R}^n</math> is the dual unit ball of <math> \|-\| </math> (the unit ball of the dual norm <math> \|-\|^* </math>).
Symplectic (coordinate-free) definition
The Holmes–Thompson volume can be defined without coordinates: if <math> A\subseteq V</math> is a measurable set in an n-dimensional real normed space <math>(V,\|-\|),</math> then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form <math> \frac 1{n!}\overbrace{\omega\wedge\cdots\wedge\omega}^n</math> over the set <math> A\times B^* </math>,
- <math>\operatorname{Vol}_{HT}(A)=\left|\int_{A\times B^*}\frac1{n!}\omega^n\right|</math>
where <math> \omega </math> is the standard symplectic form on the vector space <math> V\times V^* </math> and <math> B^*\subseteq V^*</math> is the dual unit ball of <math>\|-\|</math>.
This definition is consistent with the previous one, because if each point <math> x\in V </math> is given linear coordinates <math> (x_i)_{0\leq i<n} </math> and each covector <math> \xi \in V^* </math> is given the dual coordinates <math> (xi_i)_{0\leq i<n} </math> (so that <math> \xi(x)=\sum_i \xi_i x_i </math>), then the standard symplectic form is <math> \omega=\sum_i \mathrm d x_i \wedge \mathrm d \xi_i </math>, and the volume form is
- <math> \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},</math>
whose integral over the set <math> A\times B^* \subseteq V\times V^* \cong \mathbb R^n \times \mathbb R^n </math> is just the usual volume of the set in the coordinate space <math> \mathbb R ^{2n} </math>.
Volume in Finsler manifolds
More generally, the Holmes–Thompson volume of a measurable set <math> A </math> in a Finsler manifold <math> (M,F) </math> can be defined as
- <math>\operatorname{Vol}_\text{HT}(A):=\int_{B^*A} \frac 1{n!} \omega ^n,</math>
where <math> B^*A=\{(x,p)\in \mathrm T^*M:\ x\in A\text{ and }\xi\in \mathrm T^*_xM\text{ with }\|\xi\|_x^*\leq 1\}</math> and <math>\omega </math> is the standard symplectic form on the cotangent bundle <math> \mathrm T^*M </math>. 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 <math> M </math> is a region in coordinate space <math> \mathbb R^n </math>, then the tangent and cotangent spaces at each point <math> x\in M </math> can both be identified with <math> \mathbb R^n </math>. The Finsler metric is a continuous function <math> F:TM=M\times\mathbb R^n \to [0,+\infty) </math> that yields a (possibly asymmetric) norm <math> F_x:v \in \mathbb R^n\mapsto \|v\|_x=F(x,v)</math> for each point <math> x\in M </math>. The Holmes–Thompson volume of a subset Шаблон:Math can be computed as
- <math> \operatorname{Vol}_{\textrm{HT}}(A) = |B^*A| = \int_A |B^*_x| \,\mathrm d\operatorname{Vol_n}(x) </math>
where for each point <math> x\in M </math>, the set <math> B^*_x \subseteq \mathbb R^n </math> is the dual unit ball of <math> F_x </math> (the unit ball of the dual norm <math> F_x^* = \|-\|_x^* </math>), the bars <math> |-| </math> denote the usual volume of a subset in coordinate space, and <math> \mathrm d\operatorname{Vol_n}(x) </math> is the product of all Шаблон:Math coordinate differentials <math> \mathrm dx_i </math>.
This formula follows, again, from the fact that the Шаблон:Math-form <math>\textstyle{ \frac 1{n!} \omega ^n }</math> is equal (up to a sign) to the product of the differentials of all <math> n </math> coordinates <math> \mathrm x_i </math> and their dual coordinates <math> \xi_i </math>. The Holmes–Thompson volume of Шаблон:Math is then equal to the usual volume of the subset <math> B^*A = \{(x,\xi)\in M\times \mathbb R^n : \xi\in B^*_x \} </math> of <math> \mathbb R^{2n} </math>.
Santaló's formula
If <math> A </math> is a simple region in a Finsler manifold (that is, a region homeomorphic to a ball, with convex boundary and a unique geodesic along <math> A </math> joining each pair of points of <math> A </math>), then its Holmes–Thompson volume can be computed in terms of the path-length distance (along <math> A </math>) between the boundary points of <math> A </math> 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 <math>(\mathbb{R}^n,\|-\|_2)</math>. 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
Шаблон:Cite book Шаблон:Reflist
