In geometry of
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s, the Holmes–Thompson volume is a notion of
volume
Volume is a measure of occupied 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). The de ...
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
of a measurable set
in a normed space
is defined as the 2''n''-dimensional
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
of the product set
where
is the dual unit ball of
(the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
of the
dual norm
In functional analysis, the dual norm is a measure of size for a 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 dual space. The dual n ...
).
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 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 the ...
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 point
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
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curv ...
can be defined as
::
where
and
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 may ...
. 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 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 connection. ...
(shortest curves) contained in it (such as
systolic inequalities and
filling volumes) because, according to
Liouville's theorem, the
geodesic flow
In geometry, a geodesic () is a curve representing in some sense the 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 connection. ...
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