Cauchy’s Surface Area Formula in the Funk Geometry

Cauchy’s surface area formula is a surprising and elegant result in Euclidean geometry. It states that the surface area of any convex body is a constant multiple of the average $(d-1)$-dimensional measure of its orthogonal projections, or “shadows”, taken over all possible directions. To state the result formally in any dimension $d\ge 1$, let $S^d$ denote the $d$-dimensional unit sphere, ${\lambda }_d$ denote the standard $d$-dimensional Lebesgue measure, ${\omega }_d$ denote the volume of a $d$-dimensional Euclidean unit ball, and $(\cdot |\cdot )$ denote orthogonal projection. Cauchy’s formula states that for any convex body $K$ in ${ℝ}^d$, its surface area is

where $u^{⟂}$ is the linear subspace orthogonal to $u$. The related Crofton formula allows us to express the surface area of a convex body $K$ as the average number of lines that intersect $K$, assuming that the lines are sampled from an appropriate measure.

Cauchy’s formula is computationally fundamental, as it underpins algorithms in applications where estimating global geometric properties from lower-dimensional measurements is a recurring task, including geometric tomography [18], stereology [10], and surface area estimation for digitized 3D objects [24, 22]. Integral formulas of this form are the essential first steps in efficient sampling processes, such as computing $\epsilon$-nets [3, 14], since they implicitly define the probability distribution upon which to base the sampling.

Modern applications have increasingly explored non-Euclidean spaces, such as the Hilbert geometry and its close relative, the Funk geometry. (See Section 2 for definitions.) These geometries arise naturally whenever the domain of interest is a convex set. Examples include the analysis of discrete probability distributions in machine learning, where the domain is the probability simplex [12, 25], analysis of networks through hyperbolic geometry [21], analysis of positive definite matrices in deep learning [20, 23], and lattice-based cryptosystems [8]. Hilbert is more widely studied than Funk, but Faifman argues that for many applications, including computing volumes and areas, the Funk geometry is more natural and yields cleaner results [16].

While the metric properties of these spaces are well-understood in differential geometry [16, 26, 27], computational aspects of these geometries are only beginning to emerge. Recent work has employed the Hilbert geometry for polytope approximation [7, 8], clustering [25], and approximate membership queries [2]. Despite this growing algorithmic interest, the integral geometry of these spaces lacks the computational primitives required for efficient implementations. A major barrier is that the standard definition of surface area (the Holmes–Thompson area [19]) relies on symplectic forms or global integrals involving the polar body, making direct numerical evaluation prohibitively complex for high-dimensional or large-scale settings.

In this paper, we bridge this gap by establishing explicit Cauchy and Crofton formulas for the Funk geometry. Our formulas transform the abstract definitions of Finsler area into simple, concrete geometric quantities, which are readily accessible to discrete computation. Beyond the specific interest in Funk geometry, our results reveal it as a unifying bridge connecting diverse geometric settings.

Our main result is a Funk analog of Cauchy’s formula. Let $K$ be a convex body in ${ℝ}^d$, and let $G$ be a convex set contained in the interior of $K$. Our objective is to compute the Holmes–Thompson surface area of $G$ in the Funk geometry induced by $K$, which we denote by ${\mathrm{area}}_K^F(G)$. For simplicity, let us assume for now that $K$’s boundary is strictly convex, implying that for each direction $u\in S^{d-1}$, there is a unique boundary point $v_K(u)$ whose supporting hyperplane is orthogonal to $u$ (see Figure 1(a)). Define the central shadow $S_K(G,u)$ as the slice of the cone subtended by $G$ at $v_K(u)$, orthogonal to the direction $u$:

where $-u^{\ast }$ is the hyperplane tangent to $S^{d-1}$ at $-u$ and $\mathrm{cone}(G-v_K(u))$ is the translation of the subtended cone to the origin (see Figure 1(b)). The measure of this section captures the “visual size” of $G$ as seen from the boundary point $v_K(u)$. Just as the Euclidean formula averages orthogonal shadows, our Funk formula averages these central shadows. We show that $G$’s surface area is proportional to the average area of these shadows, taken over all directions $u$.

This offers a “tomographic interpretation” in the sense that the intrinsic Funk surface area can be recovered solely from these central slices, which capture the visibility of $G$ from the boundary of $K$. It is interesting to note that, while the Holmes–Thompson surface area of $G$ (presented in Section 2.2) is defined in terms of the areas of the polars of Finsler balls in the Funk geometry induced by $K$, the formula of Theorem 1.1 involves only simple Euclidean quantities and the standard Lebesgue measure.

Our formulation connects the Funk geometry to several classical settings. When $K$ is expanded to infinity, the central shadows become parallel projections, and our result converges to a Cauchy-type formula for the surface area in Minkowski spaces. In the special case where $K$ is a Euclidean ball, this limit yields the classical Euclidean Cauchy formula. Furthermore, for a finite Euclidean ball $K$, our formula yields the exact surface area in the Beltrami-Klein model of hyperbolic geometry [13], generalizing planar observations by Alexander et al. [5]. Finally, for arbitrary convex bodies, our Funk surface area serves as a constant-factor approximation for the Hilbert surface area, with factors depending only on the dimension $d$. These results have been omitted due to space restrictions, but they appear in the full version of the paper [9].

Crucially for algorithmic applications, we derive a discrete surface area formula when $K$ is a polytope. We show that the Funk surface area of a body $G$ nested within $K$ can be decomposed into a sum of local terms associated with the vertices of $K$. Each vertex $v$ of $K$ is naturally associated with a pointed cone $K_v=\mathrm{cone}(K-v)$, which is the cone subtended by $K$ after translating $v$ to the origin (see Figure 1(c)). The body $G$ is similarly associated with a cone $G_v=\mathrm{cone}(G-v)$. In Section 2.3, we define a simple shadow-based notion of the Funk volume of $G_v$ with respect to $K_v$, which we denote by ${\mathrm{vol}}_{K_v}^F(G_v)$. We show that the total surface area of $G$ can be expressed as the sum of these cone-based Funk volumes over the vertices of $K$.

While the cone-based Funk volume defined in Section 2.3 involves integration, it can be readily estimated through random sampling. Furthermore, this decomposition scales linearly with the vertex count of $K$ (see the discussion following the proof of Theorem 1.1 in Section 4). This is in contrast to prior Crofton measures for Hilbert geometries, which involve quadratic enumerations over face pairs [30].

Finally, we establish a Funk–Crofton formula (Theorem 4.4), expressing the Funk surface area as the $K$-dependent measure of the set of line parameters whose associated lines intersect the body. This result provides the basis for Monte-Carlo estimation of surface area, similar to methods used in Euclidean geometry [1, 22]. That is, one can approximate the Funk surface area simply by sampling from the associated parameter distribution and counting intersections of the corresponding lines with the body, bypassing the need for complex analytical evaluation.

Let us begin by presenting notation and terminology that will be used throughout the paper. We use $⟨\cdot ,\cdot ⟩$ for the standard inner (dot) product on ${ℝ}^d$, and $\parallel \cdot \parallel =\sqrt{⟨\cdot ,\cdot ⟩}$ for the Euclidean norm. Let $O$ denote the origin, let $B^d$ be the Euclidean unit ball centered at $O$, and let $S^{d-1}=\partial B^d$ be the unit sphere. Given a linear subspace $E\subset {ℝ}^d$ and a set $K\subset {ℝ}^d$, let $K|E$ denote the orthogonal projection of $K$ onto $E$.

A convex body in ${ℝ}^d$ is a compact convex set with nonempty interior. Given a convex set $K$, we denote its boundary and interior by $\partial K$ and $\mathrm{int}(K)$, respectively. For $\alpha \ge 0$, $\alpha K$ denotes the dilation of $K$ by factor $\alpha$ about the origin, and for $x\in {ℝ}^d$, $K+x$ denotes the translate of $K$ by $x$. Given a convex body $K$ and a direction $u\in S^{d-1}$, the support function of $K$ is $h_K(u)=sup\{⟨u,x⟩:x\in K\}$, and the corresponding supporting hyperplane is

For $1\le k\le d$, let ${\lambda }_k$ denote the $k$-dimensional Hausdorff measure on ${ℝ}^d$. In particular, ${\lambda }_d$ is the Lebesgue measure, and on any $C^1$ $k$-dimensional submanifold of ${ℝ}^d$, ${\lambda }_k$ agrees with the usual $k$-dimensional surface area. Let $\sigma$ denote the standard rotation-invariant surface measure on $S^{d-1}$. Finally, let ${\omega }_d={\lambda }_d(B^d)$.

Given two $C^1$ hypersurfaces $\mathrm{\Phi },\mathrm{\Psi }\subset {ℝ}^d$ and a differentiable map $f:\mathrm{\Phi }\to \mathrm{\Psi }$, the Jacobian of $f$ at $x\in \mathrm{\Phi }$ is the local area-stretch factor induced by the differential $Df(x)$ on the tangent space of $\mathrm{\Phi }$ at $x$. Equivalently, it is the factor appearing in the change-of-variables formula for surface integrals (see, for example, [15]).

Given two convex sets $A,B\subset {ℝ}^d$, their Minkowski sum is

Given a convex body $K\subset {ℝ}^d$, its difference body is

which is centrally symmetric (see Figure 2(a)).

For any nonempty set $K\subset {ℝ}^d$, its polar, denoted $K^{\circ }$, is defined by

(see Figure 2(b)). The polar has several standard properties (see, for example, Barvinok [11]). It is always closed, convex, and contains the origin. If $O\in \mathrm{int}(K)$, then $K^{\circ }$ is bounded. Moreover, if $K$ is a convex body with $O\in \mathrm{int}(K)$, then $K^{\circ }$ is also a convex body, and ${(K^{\circ })}^{\circ }=K$. Finally, polarity reverses inclusion: if $K_1\subseteq K_2$, then $K_2^{\circ }\subseteq K_1^{\circ }$.

For a linear subspace $E\subseteq {ℝ}^d$ and a nonempty set $G\subseteq E$, define the polar of $G$ in $E$ by

The next lemma is the standard duality between sections and projections, stated in the form needed later. A proof may be found in [33, Theorem 2.2.9 and Corollary 2.2.10].

In this section, we establish a discrete surface area formula for the case where $K$ is a convex polytope. Our result provides a vertex-based decomposition of the Funk surface area similar in spirit to that of Alexander et al. [5] in the planar Hilbert geometry, but derived by a distinct approach that applies in arbitrary dimensions.

Our proof proceeds by decomposing the Funk surface area of $G$ into local contributions associated with the vertices of the ambient polytope $K$. For each vertex $v\in V(K)$, let $K_v=\mathrm{cone}(K-v)$ and $G_v=\mathrm{cone}(G-v)$ denote the local cones subtended at $v$. We will show that the total Funk surface area of $G$ is the sum of the Funk volumes of the cones $G_v$ relative to $K_v$. We begin with a lemma that expresses the Funk volume of $\mathrm{cone}(G)$ relative to a pointed cone $K$ as a boundary integral involving the projected areas of the dual cross-sections $F_K(x)$. In the statement below, $T_x$ denotes the tangent space of $G$ at $x$.

The next step in our derivation utilizes the dual facet characterization (Corollary 2.5) to decompose the projection of the polar body into a sum over its facets.

The proofs of Lemmas 3.1 and 3.2 appear in the full version of the paper [9].

We are now ready to prove the vertex decomposition theorem from Section 1.

We now derive the general Cauchy formula from the polytope case. The key additional ingredient is a representation of the Funk volume of a cone as an average of the areas of its central shadows. Combined with Theorem 1.2, this yields the formula for polytopal $K$, and the general case then follows by approximation.

We established an explicit Cauchy-type formula for the Holmes–Thompson surface area in the Funk geometry induced by a convex body $K\subset {ℝ}^d$. This formula expresses the surface area of a convex body $G$ as the average, over directions $u\in S^{d-1}$, of the $(d-1)$-dimensional measures of the corresponding central shadows of $G$ (Theorem 1.1). For polytopal $K$, this identity admits a discrete vertex-based decomposition (Theorem 1.2), and the same framework yields a Crofton-type representation in terms of an explicit measure on the space of unoriented lines intersecting $G$ (Theorem 4.4).

Taken together, these results provide a concrete shadow-averaging principle for surface area in a projective Finsler setting. From a computational perspective, our formulas involve Euclidean $(d-1)$-dimensional volumes of explicitly defined slices or projections, thereby avoiding direct evaluation of the Holmes–Thompson definitions through polars of pointwise Finsler balls. In the polytopal case, the resulting decomposition is especially simple, replacing more combinatorial Crofton descriptions based on pairs of faces by a sum over vertices and normal cones.

Our work raises several natural open problems. A first direction is to develop provably efficient randomized algorithms for estimating Funk surface area from the Cauchy and Crofton formulas, ideally with explicit variance bounds and high-probability guarantees. For polytopal $K$, this includes efficient sampling from the spherical normal-cone decomposition and efficient evaluation or approximation of the associated central shadows. It would also be interesting to determine to what extent these formulas can serve as practical primitives for geometric computation in Funk- and Hilbert-type domains, in a manner analogous to the role of Cauchy and Crofton formulas in Euclidean stereology, tomography, and randomized surface area estimation.

A second direction is to investigate whether this shadow-based approach extends beyond surface area to higher-order intrinsic quantities, such as the Holmes–Thompson analogs of quermassintegrals or curvature measures. A third is to better understand the relationship with Hilbert geometry, especially by identifying conditions on $K$ under which the Funk formula yields the exact Hilbert surface area beyond the known cases of $d=2$ and ellipsoids. More generally, it would be interesting to determine how geometric properties of $K$ govern the approximation gap between Hilbert and Funk surface areas.