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
of a measurable set
in a normed space
is defined as the 2''n''-dimensional
measure of the product set
where
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
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 t ...
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 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 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
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