Busemann Volume in Finsler Geometry

• Busemann volume is the Finsler geometric volume form defined from the volumes of tangent space unit balls, serving as a canonical density for positive-definite manifolds.
• In Lorentzian Finsler spacetimes, the classical construction fails due to noncompact unit balls and degeneracies, prompting the use of compact ellipsoids for volume definition.
• A privileged time orientation is used to derive a positive-definite osculating metric, ensuring a canonical minimal Riemannian volume form even in challenging Lorentzian contexts.

Busemann volume, in the sense relevant to Finsler geometry, is the Busemann–Hausdorff-type volume form obtained from the volumes of unit balls in tangent spaces. In the positive-definite Finsler setting this is the classical Busemann–Hausdorff construction, while in time orientable Finsler spacetimes the direct Lorentzian analogue fails because the usual unit balls and indicatrices are noncompact. A precise spacetime replacement was proposed by defining a positive-definite metric from a time orientation, replacing the noncompact indicatrix by a compact ellipsoid, and then using the resulting ellipsoidal volumes to define the minimal Riemannian volume form, interpreted as the Lorentzian spacetime analogue of the Busemann–Hausdorff volume (Voicu, 2015).

1. Classical Busemann–Hausdorff construction

For a smooth positive-definite Finsler manifold $(M,L)$, with Finsler norm

$F=\sqrt{|L|},$

the closed Finsler unit ball at $x\in M$, in a positively oriented basis of $T_xM$, is

$$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$

In the positive-definite case $B_x$ is compact and convex (Voicu, 2015).

The classical Busemann–Hausdorff volume form is

$$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$

where $B\subset \mathbb{R}^n$ is the Euclidean unit ball, $\mathrm{Vol}(B)$ is its Euclidean volume, and $\mathrm{Vol}(B_x)$ is the Euclidean volume of the Finsler unit ball $F=\sqrt{|L|},$0 (Voicu, 2015). The normalization is therefore entirely tangent-space based: the density is the Euclidean unit-ball volume divided by the Euclidean volume of the Finsler unit ball at the point.

A related identity used in the classical theory connects integrals over $F=\sqrt{|L|},$1 and its boundary. For a $F=\sqrt{|L|},$2 function $F=\sqrt{|L|},$3 homogeneous of degree $F=\sqrt{|L|},$4 in $F=\sqrt{|L|},$5,

$F=\sqrt{|L|},$6

with $F=\sqrt{|L|},$7 the Euclidean volume form on $F=\sqrt{|L|},$8 (Voicu, 2015). In positive-definite Finsler geometry, this formula is meaningful because the unit ball and indicatrix are compact.

2. Lorentzian obstruction and the Finsler-spacetime setting

The Lorentzian or Finsler-spacetime case is formulated in a Beem-type $F=\sqrt{|L|},$9-homogeneous framework. Let $x\in M$0 be a connected, orientable, $x\in M$1-smooth manifold of dimension $x\in M$2, let

$x\in M$3

and let $x\in M$4 be a nonempty open submanifold such that $x\in M$5, $x\in M$6, and each fiber $x\in M$7 is a positive conic set (Voicu, 2015). A smooth function

$x\in M$8

defines a pseudo-Finsler structure if $x\in M$9 for all $T_xM$0, and the fiber Hessian

$T_xM$1

has fixed signature $T_xM$2; the case $T_xM$3 is a Lorentz-Finsler space or Finsler spacetime (Voicu, 2015).

For Lorentz-Finsler spaces, admissible vectors are classified by the sign of $T_xM$4: timelike if $T_xM$5, lightlike if $T_xM$6, and spacelike if $T_xM$7. The timelike domain is

$T_xM$8

A time orientation is a smooth section

$T_xM$9

and a Finsler spacetime is time orientable if such a section exists (Voicu, 2015).

The classical Busemann–Hausdorff definition fails in Lorentzian signature for two reasons. First, the Lorentzian unit balls and indicatrices are noncompact. In Minkowski space with metric $$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$0, the set

$$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$1

is the interior of a hyperboloid, hence noncompact, so the Euclidean volumes entering the usual Busemann–Hausdorff normalization diverge (Voicu, 2015). Second, in many Lorentz-Finsler models used in applications—Randers-type, $$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$2-th root, Bogoslovsky-type, and related examples—the metric tensor $$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$3 may be undefined or degenerate along entire tangent directions (Voicu, 2015). A direct transfer of the positive-definite construction is therefore not available.

3. Ellipsoidal replacement and the minimal Riemannian volume form

The spacetime replacement begins with a time orientation $$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$4. Its associated osculating metric is

$$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$5

From this Lorentzian metric one defines a positive-definite metric by normalizing

$$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$6

and setting

$$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$7

In coordinates,

$$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$8

For any time orientation $$B_x=\{(y^i)\in \mathbb{R}^n\mid F(x,y)\le 1\}. \tag{7}$$9, $B_x$0 is positive definite and

$B_x$1

These are the key structural facts behind the volume construction (Voicu, 2015).

The compact replacement for the noncompact Lorentzian unit ball is the ellipsoid

$B_x$2

If $B_x$3 is the Euclidean unit ball, then $B_x$4 is the image of $B_x$5 under a linear map with Jacobian

$B_x$6

and hence

$B_x$7

The Euclidean volume of the ellipsoid now plays the role that $B_x$8 played in the classical Busemann–Hausdorff formula (Voicu, 2015).

Under a suitable choice of time orientation, the resulting Busemann–Hausdorff-type spacetime form is the minimal Riemannian volume form

$B_x$9

where $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$0 is a privileged time orientation. Its local density is

$$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$1

The construction is explicitly interpreted as the spacetime analogue of the Busemann–Hausdorff volume (Voicu, 2015).

4. Privileged time orientations and canonicality

Because the ellipsoids $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$2 depend on the chosen time orientation $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$3, the construction is canonical only after selecting a preferred section. For a compact domain $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$4, one considers

$$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$5

A privileged time orientation is any time orientation $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$6 such that $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$7 is a minimum of the set of critical values of $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$8, for any compact domain $$dV_{BH}=\sigma_{BH}(x)\,\theta^0\wedge \cdots \wedge \theta^{n-1}, \qquad \sigma_{BH}(x)=\frac{\mathrm{Vol}(B)}{\mathrm{Vol}(B_x)}, \tag{9}$$9 (Voicu, 2015).

The Euler–Lagrange equations are unusually simple because the density depends only on $B\subset \mathbb{R}^n$0, not on derivatives of $B\subset \mathbb{R}^n$1: $B\subset \mathbb{R}^n$2 Using the Cartan form $B\subset \mathbb{R}^n$3, defined by

$B\subset \mathbb{R}^n$4

together with

$B\subset \mathbb{R}^n$5

criticality is equivalent to

$B\subset \mathbb{R}^n$6

Hence, if $B\subset \mathbb{R}^n$7 is a critical point of $B\subset \mathbb{R}^n$8, then

$B\subset \mathbb{R}^n$9

The Cartan form therefore provides the pointwise criterion for privileged orientations (Voicu, 2015).

The resulting volume form is independent of the choice of privileged orientation. If $\mathrm{Vol}(B)$0 and $\mathrm{Vol}(B)$1 are both privileged, then

$\mathrm{Vol}(B)$2

and therefore

$\mathrm{Vol}(B)$3

This is the canonicality statement for the minimal Riemannian volume form (Voicu, 2015).

A limitation remains explicit: existence of privileged time orientations is assumed, not proved in full generality (Voicu, 2015). A plausible implication is that the construction is best viewed as a conditional extension of Busemann–Hausdorff volume to the Lorentz-Finsler setting.

5. Comparison with Holmes–Thompson-type constructions and model examples

The same framework also permits a Holmes–Thompson-type definition, but with materially stronger regularity assumptions. The paper requires that $\mathrm{Vol}(B)$4 admit a continuous extension to $\mathrm{Vol}(B)$5, and that this extension be smooth and nowhere zero. Under those hypotheses,

$\mathrm{Vol}(B)$6

defines the corresponding density (Voicu, 2015). By contrast, the Busemann–Hausdorff-type form $\mathrm{Vol}(B)$7 only needs $\mathrm{Vol}(B)$8 evaluated along the privileged timelike section $\mathrm{Vol}(B)$9, which is why it remains available for many Lorentz-Finsler metrics whose fundamental tensor is singular or undefined in parts of $\mathrm{Vol}(B_x)$0 (Voicu, 2015).

Construction	Density formula	Regularity requirement
Busemann–Hausdorff-type	$\mathrm{Vol}(B_x)$1	Works even if $\mathrm{Vol}(B_x)$2 is not defined or is degenerate along some directions
Holmes–Thompson-type	$\mathrm{Vol}(B_x)$3	Requires $\mathrm{Vol}(B_x)$4 to extend continuously to $\mathrm{Vol}(B_x)$5, smoothly and nowhere zero

Several examples in the paper clarify the distinction. For the sign-adjusted quartic Berwald–Moor structure

$\mathrm{Vol}(B_x)$6

one has

$\mathrm{Vol}(B_x)$7

every time orientation is privileged, and

$\mathrm{Vol}(B_x)$8

In this example the Holmes–Thompson-type form exists as well, and

$\mathrm{Vol}(B_x)$9

(Voicu, 2015).

For the Bogoslovsky-type toy model

$F=\sqrt{|L|},$00

the determinant

$F=\sqrt{|L|},$01

blows up along the lightlike directions $F=\sqrt{|L|},$02, so the Holmes–Thompson construction fails, but the Busemann–Hausdorff-type form still exists. The privileged time orientation is $F=\sqrt{|L|},$03, $F=\sqrt{|L|},$04, with

$F=\sqrt{|L|},$05

and therefore

$F=\sqrt{|L|},$06

(Voicu, 2015). This example isolates the main advantage of the minimal Riemannian construction.

6. Broader mathematical landscape and terminological distinctions

The expression “Busemann volume” is easily conflated with several distinct constructions that share the name “Busemann” but are not Busemann–Hausdorff volume forms. In convex geometry, many papers concern Busemann–Petty type volume comparison or Busemann intersection inequalities, where the problem is whether section-volume data control total volume. Examples include comparison by concurrent cross-sections of complex lines (Grinberg, 2017), the complex hyperbolic Busemann–Petty problem (Dann, 2011), its lower-dimensional complex-hyperbolic analogue (Dann, 2013), stability for complex convex bodies (Koldobsky, 2011), and the entropic extension to even log-concave functions (Fang et al., 2020). These are volume-comparison theorems, not canonical volume forms.

A second nearby meaning appears in affine convex geometry. The $F=\sqrt{|L|},$07-Busemann random simplex inequality is reformulated through the Euclidean volume of the polar of a canonically associated convex body $F=\sqrt{|L|},$08, and the $F=\sqrt{|L|},$09 functional Busemann–Petty centroid inequality treats moment bodies and functional mixed volumes (Haddad, 2019, Haddad et al., 2019). Here again the underlying volume is ordinary Euclidean volume, not a Busemann–Hausdorff density.

A third use of the name comes from Busemann functions. In harmonic Hadamard and Damek–Ricci settings, positivity of the Hessian of Busemann functions is linked to rank one, visibility, and purely exponential volume growth (Itoh et al., 2017). In asymptotically harmonic manifolds, Busemann functions generate volume-preserving maps and yield uniform bounds for the volume of intersections of horospheres (Kim et al., 2022). These results concern the geometric and dynamical consequences of Busemann functions rather than a volume form called Busemann volume.

Finally, in integrable probability, Busemann functions and the Busemann process describe asymptotic last-passage increments in corner growth and exponential last-passage percolation (Fan et al., 2018, Bates et al., 14 Jun 2025). Those objects are probabilistic cocycles and have no direct connection to Busemann–Hausdorff volume.

Within this wider landscape, the specifically Finslerian sense of Busemann volume is therefore the Busemann–Hausdorff-type density, and in time orientable Finsler spacetimes its technically viable realization is the minimal Riemannian volume form built from compact ellipsoids associated to privileged time orientations (Voicu, 2015).