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 of a measurable set in a normed space is defined as the 2n-dimensional measure of the product set where is the dual unit ball of (the unit ball of the dual norm ). 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 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 . 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 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 that yields a (possibly asymmetric) norm for each point . The Holmes–Thompson volume of a subset can be computed as where for each point , the set is the dual unit ball of (the unit ball of the dual norm ), the bars denote the usual volume of a subset in coordinate space, and is the product of all coordinate differentials . This formula follows, again, from the fact that the -form is equal (up to a sign) to the product of the differentials of all coordinates and their