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In geometry of
normed space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
s, the Holmes–Thompson volume is a notion of
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
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.


Definition

The Holmes–Thompson volume \operatorname_\text(A) of a measurable set A\subseteq R^n in a normed space (\mathbb^n,\, -\, ) is defined as the 2''n''-dimensional measure of the product set A\times B^*, where B^* \subseteq \mathbb^n is the dual unit ball of \, -\, (the
unit ball Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
of the
dual norm In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space. Definition Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
\, -\, ^* ).


Symplectic (coordinate-free) definition

The Holmes–Thompson volume can be defined without coordinates: if A\subseteq V is a measurable set in an ''n''-dimensional real normed space (V,\, -\, ), then its Holmes–Thompson volume is defined as the absolute value of the integral of the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
\frac 1\overbrace^n over the set A\times B^* , :\operatorname_(A)=\left, \int_\frac1\omega^n\ where \omega is the standard symplectic form on the vector space V\times V^* and B^*\subseteq V^* is the dual unit ball of \, -\, . This definition is consistent with the previous one, because if each point x\in V is given linear coordinates (x_i)_ and each covector \xi \in V^* is given the dual coordinates (xi_i)_ (so that \xi(x)=\sum_i \xi_i x_i ), then the standard symplectic form is \omega=\sum_i \mathrm d x_i \wedge \mathrm d \xi_i , and the volume form is : \frac 1 \omega^n = \pm\; \mathrm d x_0 \wedge \dots \wedge \mathrm d x_ \wedge \mathrm d \xi_0 \wedge \dots \wedge \mathrm d \xi_, whose integral over the set 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 \mathbb R ^ .


Volume in Finsler manifolds

More generally, the Holmes–Thompson volume of a measurable set A in a Finsler manifold (M,F) can be defined as ::\operatorname_\text(A):=\int_ \frac 1 \omega ^n, where B^*A=\ and \omega is the standard symplectic form on the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
\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 In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
(shortest curves) contained in it (such as systolic inequalities and filling volumes) because, according to Liouville's theorem, the geodesic flow preserves the symplectic volume of sets in the cotangent bundle.


Computation using coordinates

If M is a region in coordinate space \mathbb R^n , then the tangent and cotangent spaces at each point x\in M can both be identified with \mathbb R^n . The Finsler metric is a continuous function F:TM=M\times\mathbb R^n \to (possibly asymmetric) norm F_x:v \in \mathbb R^n\mapsto \, v\, _x=F(x,v) for each point x\in M . The Holmes–Thompson volume of a subset can be computed as :: \operatorname_(A) = , B^*A, = \int_A , B^*_x, \,\mathrm d\operatorname(x) where for each point x\in M , the set B^*_x \subseteq \mathbb R^n is the dual unit ball of F_x (the unit ball of the
dual norm In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space. Definition Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
F_x^* = \, -\, _x^* ), the bars , -, denote the usual volume of a subset in coordinate space, and \mathrm d\operatorname(x) is the product of all coordinate differentials \mathrm dx_i . This formula follows, again, from the fact that the -form \textstyle is equal (up to a sign) to the product of the differentials of all n coordinates \mathrm x_i and their dual coordinates \xi_i . The Holmes–Thompson volume of is then equal to the usual volume of the subset B^*A = \ of \mathbb R^ .


Santaló's formula

If 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 A joining each pair of points of A ), then its Holmes–Thompson volume can be computed in terms of the path-length distance (along A ) between the boundary points of A using Santaló's formula, which in turn is based on the fact that the geodesic flow on the cotangent bundle is Hamiltonian dynamics, Hamiltonian.


Normalization and comparison with Euclidean and Hausdorff measure

The original authors used 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 (\mathbb^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 In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assi ...
. 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

{{DEFAULTSORT:Holmes-Thompson volume Normed spaces Differential geometry Measure theory Finsler geometry Integral geometry Systolic geometry