Holmes–Thompson volume#Normalization and comparison with Euclidean and Hausdorff measure

The Holmes–Thompson volume $\backslash operatorname\{Vol\}\_\backslash text\{HT\}(A)$ of a measurable set $A\backslash subseteq\; R^n$ in a normed space $(\backslash mathbb\{R\}^n,\backslash |-\backslash |)$ is defined as the 2n-dimensional measure of the product set $A\backslash times\; B^*,$ where $B^*\; \backslash subseteq\; \backslash mathbb\{R\}^n$ is the dual unit ball of $\backslash |-\backslash |$ (the unit ball of the dual norm $\backslash |-\backslash |^*$).

The Holmes–Thompson volume can be defined without coordinates: if $A\backslash subseteq\; V$ is a measurable set in an n-dimensional real normed space $(V,\backslash |-\backslash |),$ then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form $\backslash frac\; 1\{n!\}\backslash overbrace\{\backslash omega\backslash wedge\backslash cdots\backslash wedge\backslash omega\}^n$ over the set $A\backslash times\; B^*$,

:$\backslash operatorname\{Vol\}\_\{HT\}(A)=\backslash left|\backslash int\_\{A\backslash times\; B^*\}\backslash frac1\{n!\}\backslash omega^n\backslash right|$

where $\backslash omega$ is the standard symplectic form on the vector space $V\backslash times\; V^*$ and $B^*\backslash subseteq\; V^*$ is the dual unit ball of $\backslash |-\backslash |$.

This definition is consistent with the previous one, because if each point $x\backslash in\; V$ is given linear coordinates

:$\backslash frac\; 1\{n!\}\; \backslash omega^n\; =\; \backslash pm\backslash ;\; \backslash mathrm\; d\; x\_0\; \backslash wedge\; \backslash dots\; \backslash wedge\; \backslash mathrm\; d\; x\_\{n-1\}\; \backslash wedge\; \backslash mathrm\; d\; \backslash xi\_0\; \backslash wedge\; \backslash dots\; \backslash wedge\; \backslash mathrm\; d\; \backslash xi\_\{n-1\},$

whose integral over the set $A\backslash times\; B^*\; \backslash subseteq\; V\backslash times\; V^*\; \backslash cong\; \backslash mathbb\; R^n\; \backslash times\; \backslash mathbb\; R^n$ is just the usual volume of the set in the coordinate space $\backslash mathbb\; R\; ^\{2n\}$.

More generally, the Holmes–Thompson volume of a measurable set $A$ in a Finsler manifold $(M,F)$ can be defined as

::$\backslash operatorname\{Vol\}\_\backslash text\{HT\}(A):=\backslash int\_\{B^*A\}\; \backslash frac\; 1\{n!\}\; \backslash omega\; ^n,$

where $B^*A=\backslash \{(x,p)\backslash in\; \backslash mathrm\; T^*M:\backslash \; x\backslash in\; A\backslash text\{\; and\; \}\backslash xi\backslash in\; \backslash mathrm\; T^*\_xM\backslash text\{\; with\; \}\backslash |\backslash xi\backslash |\_x^*\backslash leq\; 1\backslash \}$ and $\backslash omega$ is the standard symplectic form on the cotangent bundle $\backslash mathrm\; T^*M$. 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{{cite journal |title=Local extremality of the Calabi–Croke sphere for the length of the shortest closed geodesic |last=Sabourau |first=Stéphane |journal=Journal of the London Mathematical Society |volume=82 |issue=3 |year=2010 |pages=549–562 |arxiv=0907.2223 |doi=10.1112/jlms/jdq045|s2cid=1156703 }}{{cite journal |title=Isosystolic inequalities for optical hypersurfaces |last1=Álvarez Paiva |first1=Juan-Carlos |last2=Balacheff |first2=Florent |last3=Tzanev |first3=Kroum |journal=Advances in Mathematics |volume=301 |year=2016 |pages=934–972 |arxiv=1308.5522 |doi=10.1016/j.aim.2016.07.003|doi-access=free|s2cid=119175687 }} and filling volumes{{cite journal |title=Volume Comparison via Boundary Distances |journal=Proceedings of ICM |year=2010 |last1=Ivanov | first1=Sergei V. |arxiv=1004.2505}}{{cite journal |title=On two-dimensional minimal fillings |language = ru |first=Sergei V. |last=Ivanov |journal=Algebra i Analiz |volume=13 |number=1 |pages=26–38 |year=2001}}{{cite journal |title=On two-dimensional minimal fillings |language=en | first = Sergei V. |last=Ivanov |journal=St. Petersburg Math. J. |volume=13 |number=1 |year=2002 |pages=17–25 |mr=1819361}}{{ cite journal |title=Filling minimality of Finslerian 2-discs |first=Sergei V. |last=Ivanov |journal=Proc. Steklov Inst. Math. |volume=273 |number=1 |pages=176–190 |year=2011 |doi=10.1134/S0081543811040079 | arxiv = 0910.2257|s2cid=115167646 }}{{cite journal |title=Local monotonicity of Riemannian and Finsler volume with respect to boundary distances |last=Ivanov |first=Sergei V. |journal=Geometriae Dedicata |volume=164 |issue=2013 |year=2013 |pages=83–96 |arxiv=1109.4091 |doi=10.1007/s10711-012-9760-y|s2cid=119130237 }}) because, according to Liouville's theorem, the geodesic flow preserves the symplectic volume of sets in the cotangent bundle.

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

:: $\backslash operatorname\{Vol\}\_\{\backslash textrm\{HT\}\}(A)\; =\; |B^*A|\; =\; \backslash int\_A\; |B^*\_x|\; \backslash ,\backslash mathrm\; d\backslash operatorname\{Vol\_n\}(x)$

where for each point $x\backslash in\; M$, the set $B^*\_x\; \backslash subseteq\; \backslash mathbb\; R^n$ is the dual unit ball of $F\_x$ (the unit ball of the dual norm $F\_x^*\; =\; \backslash |-\backslash |\_x^*$), the bars $|-|$ denote the usual volume of a subset in coordinate space, and $\backslash mathrm\; d\backslash operatorname\{Vol\_n\}(x)$ is the product of all {{math|n}} coordinate differentials $\backslash mathrm\; dx\_i$.

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

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.

{{cite encyclopedia |title=Santaló formula |encyclopedia=Encyclopedia of Mathematics |url=http://www.encyclopediaofmath.org/index.php?title=Santal%C3%B3_formula&oldid=23516}}

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 $(\backslash mathbb\{R\}^n,\backslash |-\backslash |\_2)$. 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).

{{cite book |last1=Álvarez-Paiva |first1=Juan-Carlos |last2=Thompson |first2=Anthony C. |editor1-last=Bao |editor1-first=David | editor2-last=Bryant|editor2-first=Robert L.|editor3-last=Chern|editor3-first=Shiing-Shen|editor4-last=Shen|editor4-first=Zhongmin|title=A sampler of Riemann-Finsler geometry |publisher=Cambridge University Press |series=MSRI Publications|volume=50|year=2004 |pages=1–48 |chapter=Chapter 1: Volumes on Normed and Finsler Spaces|chapter-url=http://library.msri.org/books/Book50/files/02AT.pdf|isbn=0-521-83181-4|mr=2132656}}

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Category:Normed spaces

Category:Differential geometry

Category:Measure theory

Category:Finsler geometry

Category:Integral geometry

Category:Systolic geometry