Holmes–Thompson Volume in Finsler Geometry

• Holmes–Thompson Volume is a symplectic measure on Finsler manifolds defined via the unit co-disc bundle, contrasting with the tangent-based Busemann–Hausdorff Volume.
• It unifies geometric analysis by linking duality, Crofton formulas, and integral geometry to quantify volumes and boundary areas.
• Its framework underpins studies in convex geometry, boundary rigidity, and valuation theory, offering robust applications in both theoretical and applied mathematics.

Holmes–Thompson volume is a symplectic definition of volume for Finsler manifolds and normed spaces, constructed from the unit co-disc bundle rather than the tangent unit ball. For an $n$-dimensional Finsler manifold $(M,F)$, with dual norm $F^*$, co-unit ball bundle $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$, and canonical symplectic form $\omega=d\alpha$ on $T^*M$, it is given by

$$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$

where $\omega_n$ is the Euclidean volume of the unit ball in $\mathbb{R}^n$ (Ivanov, 2011). More generally, for a $k$-dimensional continuous Finsler manifold, the corresponding $(M,F)$0-dimensional Holmes–Thompson volume is

$(M,F)$1

(Balacheff et al., 2020). In the Riemannian case it agrees with ordinary Riemannian volume, while in genuinely Finsler settings it is distinguished by its cotangent-bundle definition, its compatibility with duality and Crofton formulas, and its central role in convex, projective, and integral geometry (Koehler, 2011).

1. Symplectic definition and normalization

A Finsler metric $(M,F)$2 on a smooth $(M,F)$3-manifold is positively homogeneous, positive away from the zero section, smooth on $(M,F)$4, and fiberwise strictly convex; non-reversible metrics are allowed (Ivanov, 2011). Its dual norm is defined fiberwise by

$(M,F)$5

and the co-unit ball $(M,F)$6 is the basic local object in the Holmes–Thompson construction (Ivanov, 2011).

The global definition is equivalent to a density formula. If $(M,F)$7 is the bundle projection and $(M,F)$8 is the characteristic function of the co-unit ball bundle, then

$(M,F)$9

(Ivanov, 2011). In continuous Finsler geometry this is often written in the equivalent form

$F^*$0

which makes explicit that Holmes–Thompson volume is the normalized symplectic volume of a co-disc bundle (Balacheff et al., 2020).

Its standard contrast is with the Busemann–Hausdorff volume. Busemann volume is defined from the tangent unit ball $F^*$1, whereas Holmes–Thompson volume is defined from the dual co-ball $F^*$2 and the symplectic measure on $F^*$3 (Koehler, 2011). Both agree with the Riemannian volume in the Riemannian case, but they differ in general Finsler settings (Koehler, 2011). A basic monotonicity property is immediate from the co-ball definition: if $F^*$4 pointwise on $F^*$5, then $F^*$6, hence

$F^*$7

(Ivanov, 2011).

2. Normed spaces, hypersurface area, and convex duality

In a finite-dimensional normed space $F^*$8 with unit ball $F^*$9, Holmes–Thompson volume becomes translation invariant. For a compact set $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$0,

$B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$1

and in Euclidean coordinates, identifying dual with polar,

$B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$2

(Balacheff et al., 2020). Equivalently, in a $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$3-dimensional Minkowski space with Euclidean Lebesgue measure $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$4,

$B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$5

so the density is constant and determined by the volume of the dual unit ball (Lángi et al., 15 Jul 2025).

The same framework yields a hypersurface measure. If $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$6 is a smooth hypersurface, the induced Finsler metric gives the Holmes–Thompson boundary area

$B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$7

(Balacheff et al., 2020). For convex bodies this extends to $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$8 by approximation, since singular boundary points have Hausdorff $B^*M=\{(x,p)\in T^*M:F^*(x,p)\le 1\}$9-measure zero (Balacheff et al., 2020). A basic duality principle, due to Holmes and Thompson, is

$\omega=d\alpha$0

which makes the dual body $\omega=d\alpha$1 intrinsic to the area theory as well as to the volume theory (Balacheff et al., 2020).

This duality underlies the Holmes–Thompson boundary-area analogue of the Santaló point. For a convex body $\omega=d\alpha$2, the relevant functional is

$\omega=d\alpha$3

A minimizing point is the Santaló point of $\omega=d\alpha$4 for the Holmes–Thompson area, denoted $\omega=d\alpha$5 (Balacheff et al., 2020). If the unit ball $\omega=d\alpha$6 is of class $\omega=d\alpha$7, this functional is strictly convex and proper on $\omega=d\alpha$8, hence has a unique minimizer. In the special case $\omega=d\alpha$9, uniqueness holds in every finite-dimensional normed space, without assuming $T^*M$0 regularity (Balacheff et al., 2020).

When $T^*M$1 is smooth and positively curved, the first variation is expressed by a dual centroid-averaging operator

$T^*M$2

where the inner integral is the centroid of the projected dual body $T^*M$3 (Balacheff et al., 2020). The variation formula is

$T^*M$4

so $T^*M$5 if and only if $T^*M$6 (Balacheff et al., 2020). This gives the Holmes–Thompson area Santaló point a variational interpretation parallel to the classical centroid condition for the polar-volume Santaló point.

3. Integral geometry and Crofton theory

Holmes–Thompson volume is unusually well adapted to integral geometry. In a Minkowski space, the space of geodesics is the affine Grassmannian of lines, and it carries a symplectic form obtained by symplectic reduction from the unit sphere bundle via the Legendre map (Liu, 2010). If $T^*M$7 is the projection sending a unit tangent vector to its oriented line, then there is a unique symplectic form $T^*M$8 on the space of lines such that

$T^*M$9

(Liu, 2010). This is the symplectic origin of the Crofton measures associated with Holmes–Thompson geometry.

Álvarez Paiva’s Crofton formula gives the $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$0-dimensional Holmes–Thompson volume of a $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$1-dimensional submanifold $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$2 as

$$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$3

where $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$4 is the Crofton measure obtained by intersection pushforward from a hyperplane measure (Liu, 2010). For hypersurfaces this specializes to

$$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$5

(Liu, 2010). In Minkowski spaces this is the integral-geometric expression behind the Holmes–Thompson boundary area and its symplectic Crofton formulas (Balacheff et al., 2020).

In the plane the formulas become fully explicit. If oriented lines are parametrized by Euclidean normal angle $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$6 and signed distance $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$7, there is an even function $$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$8 such that

$$\operatorname{Vol}_{HT}(M,F)=\frac{1}{\omega_n}\int_{B^*M}\frac{\omega^n}{n!},$$9

and for a rectifiable curve $\omega_n$0,

$\omega_n$1

(Liu, 2010). The planar Holmes–Thompson area is then expressed via pairs of lines: $\omega_n$2 with $\omega_n$3 on the line space (Liu, 2010). This explicit two-line formula is a characteristic feature of the Holmes–Thompson theory and has no analogue of comparable simplicity for general Finsler volume notions.

4. Boundary distances, Santaló-type identities, and monotonicity

On simple Finsler manifolds, Holmes–Thompson volume can be reconstructed from boundary data. A simple metric on the disc has strictly convex boundary, unique minimizing geodesics, and no conjugate points; in that setting an enveloping function $\omega_n$4 parametrizes the unit co-spheres and determines the boundary distance function (Ivanov, 2011). A key lemma states that $\omega_n$5 is uniquely determined by the restriction $\omega_n$6, and hence by the boundary distance function in the simple case (Ivanov, 2011).

The resulting boundary integral identity is of Santaló type. For a simple Finsler $\omega_n$7-disk $\omega_n$8 with boundary distance $\omega_n$9,

$\mathbb{R}^n$0

(Koehler, 2011). This formula converts the cotangent symplectic definition into an integral over boundary distances and their mixed second derivatives.

Its first major consequence is local monotonicity with respect to boundary distance. For every simple Finsler metric $\mathbb{R}^n$1 on $\mathbb{R}^n$2, there exists a $\mathbb{R}^n$3 neighborhood $\mathbb{R}^n$4 such that, for all $\mathbb{R}^n$5,

$\mathbb{R}^n$6

(Ivanov, 2011). The proof uses enveloping functions, a diffeomorphism between unit tangent bundles matching geodesic trajectories by boundary endpoints, and the elementary monotonicity of Holmes–Thompson volume under pointwise enlargement of the metric (Ivanov, 2011).

A sharper comparison formula is available for two simple reversible Finsler disks $\mathbb{R}^n$7 and $\mathbb{R}^n$8: $\mathbb{R}^n$9 where

$k$0

(Koehler, 2011). In dimension $k$1, the kernel is automatically positive, so $k$2 implies $k$3. In dimensions $k$4 and $k$5, positivity holds under the additional assumption that $k$6 is itself a boundary distance function of a simple Finsler metric, and in all dimensions it holds under sufficient $k$7-closeness (Koehler, 2011). These results place Holmes–Thompson volume at the center of Finsler filling minimality and boundary rigidity.

5. Projective, Funk, and Hilbert geometries

In Funk geometry, Holmes–Thompson volume becomes a projectively natural measure. If $k$8 is a bounded convex domain, the forward Funk norm is

$k$9

where $(M,F)$00 is the positive time at which the ray $(M,F)$01 hits $(M,F)$02, and the forward ball of radius $(M,F)$03 about $(M,F)$04 is

$(M,F)$05

(Faifman et al., 2023). The Holmes–Thompson measure of a Borel set $(M,F)$06 is

$(M,F)$07

where $(M,F)$08 is the polar body of $(M,F)$09 with respect to $(M,F)$10 (Faifman et al., 2023). This formula makes the relation to Mahler-type quantities explicit.

A projective formulation replaces affine coordinates by the canonical invariant measure on $(M,F)$11,

$(M,F)$12

and yields

$(M,F)$13

for measurable $(M,F)$14 (Faifman, 2020). From this, Holmes–Thompson volume in Funk geometry is invariant under projective transformations and under projective polarity; the same framework also yields duality of boundary volumes and of billiard length spectra (Faifman, 2020).

For polytopal Funk geometries, the large-radius expansion of the Holmes–Thompson volume of forward balls has an explicitly combinatorial leading term: $(M,F)$15 while

$(M,F)$16

depends on the geometry as well as the combinatorics (Faifman et al., 2023). The coefficient $(M,F)$17, viewed as a function of the center $(M,F)$18, is proper and strictly convex, so it has a unique minimizer $(M,F)$19, characterized by a flag-weighted centroid condition in the dual polytope (Faifman et al., 2023). This is a large-radius analogue of the Santaló point.

In Hilbert geometry, the two-dimensional Holmes–Thompson area is

$(M,F)$20

and is projectively invariant (Flamm et al., 7 Jan 2026). The normalization is chosen so that Holmes–Thompson area agrees with hyperbolic area on ellipses; in particular, a hyperbolic ideal quadrilateral has Holmes–Thompson area $(M,F)$21 (Flamm et al., 7 Jan 2026). For standard polygons, explicit integrands are available. In the standard triangle $(M,F)$22,

$(M,F)$23

and for a positive triple of flags with triple ratio $(M,F)$24,

$(M,F)$25

(Flamm et al., 7 Jan 2026). These formulas show that Holmes–Thompson area in Hilbert geometry is computable in terms of flag invariants and projective coordinates, not merely abstractly defined.

6. Intrinsic volumes, extremal problems, and extensions

Holmes–Thompson volume also generates a valuation theory. In a finite-dimensional hypermetric normed space $(M,F)$26, the Holmes–Thompson intrinsic volumes $(M,F)$27 are the unique even, continuous, translation-invariant convex valuations of degree $(M,F)$28, normalized by

$(M,F)$29

whenever $(M,F)$30 is $(M,F)$31-dimensional and $(M,F)$32 is its affine span (Meckes, 2022). With the normalization adopted in that paper,

$(M,F)$33

while in $(M,F)$34,

$(M,F)$35

(Meckes, 2022). These intrinsic volumes satisfy a Crofton-type representation through a generating measure $(M,F)$36 on $(M,F)$37, and they enter directly into the magnitude of convex bodies: $(M,F)$38 (Meckes, 2022). Applications include a short proof of Mahler’s conjecture for zonoids and a derivation of Sudakov’s minoration inequality (Meckes, 2022).

In normed approximation theory, Holmes–Thompson volume produces extremal problems different from those for Busemann volume. For a $(M,F)$39-dimensional normed space with unit ball $(M,F)$40, the Holmes–Thompson volume of a maximal inscribed $(M,F)$41-vertex polytope $(M,F)$42 is

$(M,F)$43

and in the planar case

$(M,F)$44

for even $(M,F)$45, attained by regular origin-symmetric $(M,F)$46-gons (Lángi et al., 15 Jul 2025). The dependence on $(M,F)$47 rather than $(M,F)$48 is the defining dual feature of the Holmes–Thompson normalization (Lángi et al., 15 Jul 2025).

A further extension appears in time-orientable Finsler spacetimes. Because Lorentzian indicatrices are non-compact, the classical Holmes–Thompson construction diverges, so the spacetime theory replaces them by compact ellipsoids $(M,F)$49 associated with a privileged time orientation $(M,F)$50 and a positive-definite metric $(M,F)$51 (Voicu, 2015). Under the additional assumption that $(M,F)$52 extends smoothly and nowhere vanishingly to the slit tangent bundle, the Holmes–Thompson-type density is

$(M,F)$53

yielding a finite, coordinate-invariant spacetime volume form independent of the chosen privileged time orientation (Voicu, 2015). This suggests that the cotangent and dual-ball philosophy of Holmes–Thompson volume persists beyond the positive-definite Finsler category, although with substantial additional structure.

Across these settings, Holmes–Thompson volume functions as the symplectic measure native to Finsler and normed geometry. Its co-ball definition, duality properties, Crofton formulas, projective invariance, and compatibility with convex polarity explain why it recurs in boundary rigidity, convex projective geometry, Mahler-type inequalities, billiards, and valuation theory.