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. Definition. The Holmes–Thompson volume formula_1 of a measurable set formula_2 in a normed space formula_3 is defined as the 2"n"-dimensional measure of the product set formula_4 where formula_5 is the dual unit ball of formula_6 (the unit ball of the dual norm formula_7). Symplectic (coordinate-free) definition. The Holmes–Thompson volume can be defined without coordinates: if  formula_8 is a measurable set in an "n"-dimensional real normed space formula_9 then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form formula_10 over the set formula_11, where formula_13 is the standard symplectic form on the vector space formula_14 and formula_15 is the dual unit ball of formula_16. This definition is consistent with the previous one, because if each point formula_17 is given linear coordinates formula_18 and each covector formula_19 is given the dual coordinates formula_20 (so that formula_21), then the standard symplectic form is formula_22, and the volume form is whose  integral over the set formula_24 is just the usual volume of the set in the coordinate space formula_25. Volume in Finsler manifolds. More generally, the Holmes–Thompson volume of a measurable set formula_26 in a Finsler manifold formula_27 can be defined as where formula_29 and formula_30 is the standard symplectic form on the cotangent bundle formula_31. 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 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 formula_32 is a region in coordinate space formula_33, then the tangent and cotangent spaces at each point formula_34 can both be identified with formula_33. The Finsler metric is a continuous function formula_36 that yields a (possibly asymmetric) norm formula_37 for each point formula_34. The Holmes–Thompson volume of a subset can be computed as where for each point formula_34, the set formula_41 is the dual unit ball of formula_42 (the unit ball of the dual norm formula_43), the bars formula_44 denote the usual volume of  a subset in coordinate space, and formula_45 is the product of all coordinate differentials formula_46. This formula follows, again, from the fact that the -form formula_47 is equal (up to a sign) to the product of the differentials of all formula_48 coordinates formula_49 and their dual coordinates formula_50. The Holmes–Thompson volume of is then equal to the usual volume of the subset formula_51 of formula_52. Santaló's formula. If formula_26 is a simple region in a Finsler manifold (that is, a region homeomorphic to a ball, with convex boundary and a unique geodesic along formula_26 joining each pair of  points of formula_26), then its Holmes–Thompson volume can be computed in terms of the path-length distance (along formula_26) between the boundary points of formula_26 using Santaló's formula, which in turn is based on the fact that the geodesic flow on the cotangent bundle is 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 formula_58. 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).