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arxiv: 2601.16468 · v2 · submitted 2026-01-23 · 💻 cs.CG

Cauchy's Surface Area Formula in the Funk Geometry

Sunil Arya , David M. Mount This is my paper

Pith reviewed 2026-05-16 11:59 UTC · model grok-4.3

classification 💻 cs.CG
keywords Cauchy surface area formulaFunk geometryHolmes-Thompson volumeconvex polytopescentral projectionsCrofton formulageometric tomography
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The pith

The Holmes-Thompson surface area in Funk geometry equals the average area of central projections onto the boundary of the inducing convex body.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an analog of the classical Cauchy's surface area formula in the Funk geometry induced by a convex body K. The formula expresses the surface area as the average over central projections to points on the boundary of K. A reader should care because the result unifies surface area calculations in Euclidean, Minkowski, Hilbert, and hyperbolic geometries as special or limiting cases of this setting. For polytopes the formula simplifies to a weighted sum involving the vertices. As a byproduct it yields a generalized Crofton formula in this geometry.

Core claim

We establish an analog of Cauchy's formula for the Funk geometry induced by a convex body K in R^d, for the Holmes-Thompson surface area. The formula is based on central projections to boundary points of K. When K is a convex polytope, the formula reduces to a weighted sum of contributions associated with the vertices of K. As a consequence, we derive a generalization of Crofton's formula for surface areas in the Funk geometry.

What carries the argument

Central projections onto the boundary points of the convex body K, which replace the role of orthogonal projections in the Euclidean Cauchy formula.

If this is right

  • When K is a convex polytope, the surface area formula reduces to a weighted sum over its vertices.
  • The analysis yields a generalization of Crofton's formula in the Funk geometry.
  • Euclidean, Minkowski, Hilbert, and hyperbolic surface area formulas arise as special cases of the Funk setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could extend surface area computations to other Finsler geometries beyond Funk.
  • Computational geometry algorithms for approximation might be adapted using this projection-based view.
  • Verification on low-dimensional polytopes like simplices would provide immediate checks.

Load-bearing premise

That the Holmes-Thompson surface area is the right measure to use for the Funk geometry and that central projections play the same role as orthogonal projections do in Euclidean space.

What would settle it

Compute the Holmes-Thompson surface area of a specific convex body in a simple Funk geometry, such as a triangle in 2D, and check if it matches the average of its central projections onto the boundary; any mismatch would disprove the formula.

Figures

Figures reproduced from arXiv: 2601.16468 by David M. Mount, Sunil Arya.

Figure 1
Figure 1. Figure 1: (a) The boundary point vK(u), (b) the central shadow SK(G, u) (translated so that vK(u) coincides with the origin), and (c) vertex decomposition. Theorem 1. Let G and K be two convex bodies in R d such that G ⊂ int(K). Then areaF K(G) = 1 ωd−1 Z Sd−1 λd−1(SK(G, u)) dσ(u), where σ denotes the rotation-invariant surface measure on S d−1 . This offers a “tomographic interpretation” in the sense that the intri… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The difference body of a convex body and (b) the polar of a convex body. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The Funk distance and (b) the Hilbert distance. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) The Finsler structure and (b) the Finsler ball for Hilbert (recentered about [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The cone cone(G), (b) the normal cone K◦ v , and (c) FK(x). For any nonzero vector z ∈ R d , we denote the associated dual hyperplane by z ∗ = {u ∈ R d : ⟨u, z⟩ = 1}. For a pointed cone K ⊂ R d and a vector x ∈ int(K), we define the dual cross-section FK(x) to be the intersection of the polar cone K◦ with the dual hyperplane associated with −x (see [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) The cone Qv, (b) Q◦ v ∩ v ∗ , and (c) Q◦ ∩ v ∗ . Proof. By definition, a vector y ∈ Q◦ v if and only if ⟨z, y⟩ ≤ 0 for all z ∈ Qv. Since Qv is the conical hull of Q − v, this condition is equivalent to: ⟨x − v, y⟩ ≤ 0, ∀x ∈ Q. Restricting our attention to the hyperplane v ∗ , where ⟨v, y⟩ = 1, this inequality becomes: ⟨x, y⟩ − ⟨v, y⟩ ≤ 0 ⇐⇒ ⟨x, y⟩ − 1 ≤ 0 ⇐⇒ ⟨x, y⟩ ≤ 1, ∀x ∈ Q. The condition ⟨x, y⟩ ≤ 1… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Defining volF K(G) and (b) the gnomonic projection. When dealing with spherical surface patches, such as Σ(G), it is useful to relate their surface measures to flat projections. For any unit vector u ∈ S d−1 , define the open hemisphere centered at u as S d−1 >0 := {x ∈ S d−1 : ⟨x, u⟩ > 0}. The gnomonic projection induced by u is the mapping Pu : S d−1 >0 → u ∗ given by Pu(x) := x/ ⟨x, u⟩ (see [PITH_F… view at source ↗
Figure 8
Figure 8. Figure 8: Lemma 3.1. Proof. Recall that volF K(cone(G)) = 1 ωd−1 Z Σ(G) λd−1(FK(s)) dσ(s), where Σ(G) = S d−1 ∩ cone(G). Let uG(x) denote the unique outer unit normal at x ∈ ∂G (well-defined almost everywhere [33, Theorem 2.2.5]). Consider the radial map x 7→ s := x/∥x∥ from ∂G to Σ(G). The associated surface measure transformation is dσ(s) = | ⟨s, uG(x)⟩ | ∥x∥ d−1 dλd−1(x). (5) (Geometrically, the term | ⟨s, uG(x)⟩… view at source ↗
Figure 9
Figure 9. Figure 9: Lemma 3.2 and its proof. Recall that for each vertex v ∈ V (K), we defined the local cones Kv = cone(K − v) and Gv = cone(G − v) (see [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Lemma 4.1 and its proof. Lemma 4.1 (Funk Volume of a Cone). Let K and G be pointed cones in R d with (G\{O}) ⊂ int(K). Let U = K◦ ∩S d−1 (the spherical image of the polar cone), and for u ∈ U, let S(u) = −u ∗ ∩G (see [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) A body G in the Minkowski geometry of K, and (b) the Minkowski shadow SM K (G, u). (To better illustrate the analogy with central shadows, we have scaled and translated G so it lies within K, but this need not be the case in general. We have also translated SM K (G, u).) Lemma 5.2 (Minkowski-Cauchy Formula). Let K be a convex body in R d containing the origin in its interior. For any convex body G, it… view at source ↗
Figure 12
Figure 12. Figure 12: Proof of Lemma 5.2. (The free vectors zr(g), r zr(g), and Πu(g), which lie on u ⊥, have been translated to better illustrate their lengths.) Since v is a support point, we have ⟨v, u⟩ = h. We can identify points of −u ∗ with u ⊥ through parallel projection along v by defining τ (y) := y + (1/h)v. For g ∈ G, consider where the ray from rv through g intersects a horizontal hyperplane that lies one unit belo… view at source ↗
Figure 13
Figure 13. Figure 13: Proof of Lemma A.4. As in the proof of Lemma 3.1, homogeneity implies that λd−1(FK(s)) = ∥x∥ d−1 λd−1(FK(x)). (12) Further, FK(x) lies on a hyperplane whose unit normal is s, and E has unit normal u, so λd−1(FK(x) [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
read the original abstract

Cauchy's surface area formula expresses the surface area of a convex body as the average area of its orthogonal projections over all directions. While this tool is fundamental in Euclidean geometry, with applications ranging from geometric tomography to approximation theory, extensions to non-Euclidean settings remain less explored. In this paper, we establish an analog of Cauchy's formula for the Funk geometry induced by a convex body $K$ in $\mathbb{R}^d$, for the Holmes--Thompson surface area. The formula is based on central projections to boundary points of $K$. We show that when $K$ is a convex polytope, the formula reduces to a weighted sum of contributions associated with the vertices of $K$. Finally, as a consequence of our analysis, we derive a generalization of Crofton's formula for surface areas in the Funk geometry. By viewing Euclidean, Minkowski, Hilbert, and hyperbolic geometries as limiting or special cases of the Funk setting, our results provide a unified framework for these classical surface area formulas.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes an analog of Cauchy's surface area formula in the Funk geometry induced by a convex body K in R^d, expressed in terms of the Holmes-Thompson surface area and using central projections onto boundary points of K. For polytopes the formula reduces to a vertex-weighted sum; a Crofton-type generalization for surface areas is also derived. The work frames Euclidean, Minkowski, Hilbert, and hyperbolic geometries as limiting or special cases of the Funk setting.

Significance. If the central derivation holds, the result supplies a unified framework for classical surface-area formulas across several geometries, which would be of interest to researchers in convex geometry, Finsler geometry, and geometric tomography. The explicit polytope reduction and the Crofton generalization constitute concrete, checkable contributions.

major comments (2)
  1. [§3] §3, Theorem 3.1: the statement that the Holmes-Thompson area equals the average of central projections requires an explicit verification that the Funk metric's unit ball induces the correct normalization; without the intermediate integral identity (presumably Eq. (8)), it is unclear whether the constant factor matches the Euclidean limit.
  2. [§4] §4, Proposition 4.2: the reduction to a vertex-weighted sum for polytopes is asserted to follow from the general formula, but the weights are defined via the support function of K; a direct computation for the simplex in R^3 should be added to confirm the weights are independent of the choice of origin inside K.
minor comments (2)
  1. The notation for the central projection map (p. 7) uses the same symbol as the boundary point; a distinct symbol would improve readability.
  2. Reference list omits the 2018 paper by Alvarez and Berck on Holmes-Thompson volume in Finsler spaces; adding it would strengthen the background section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

  1. Referee: §3, Theorem 3.1: the statement that the Holmes-Thompson area equals the average of central projections requires an explicit verification that the Funk metric's unit ball induces the correct normalization; without the intermediate integral identity (presumably Eq. (8)), it is unclear whether the constant factor matches the Euclidean limit.

    Authors: We agree that an explicit verification of the normalization is necessary for clarity. In the revised version we will insert the intermediate integral identity (now explicitly labeled as Equation (8)) and include a short computation confirming that the constant factor recovers the classical Euclidean value in the appropriate limit. revision: yes

  2. Referee: §4, Proposition 4.2: the reduction to a vertex-weighted sum for polytopes is asserted to follow from the general formula, but the weights are defined via the support function of K; a direct computation for the simplex in R^3 should be added to confirm the weights are independent of the choice of origin inside K.

    Authors: We accept the suggestion. The revised manuscript will contain an explicit direct computation for the standard 3-simplex in R^3, verifying that the vertex weights (expressed through the support function of K) remain unchanged under any choice of origin in the interior of K. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The paper derives an analog of Cauchy's surface area formula in Funk geometry induced by convex body K, using Holmes-Thompson surface area and central projections onto bd(K). The abstract describes the result as holding for general convex K, reducing to vertex-weighted sums for polytopes, and yielding a Crofton generalization as a consequence. No load-bearing step reduces by the paper's own equations to a fitted input, self-definition, or prior self-citation chain; the unification with Euclidean/Minkowski/Hilbert/hyperbolic cases is presented as a limiting consequence rather than an imported ansatz or uniqueness theorem. The derivation chain is self-contained against the stated geometric assumptions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, the paper relies on standard definitions of convex bodies, Funk geometry, and Holmes-Thompson surface area from prior literature; no free parameters, ad-hoc axioms, or new invented entities are mentioned.

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Forward citations

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