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Cauchy's Surface Area Formula in the Funk Geometry

Sunil Arya, David M. Mount

TL;DR

This work develops explicit Cauchy-type formulas for Holmes-Thompson surface area in the Funk geometry induced by a convex body K, expressing area as an average of central shadows over all directions. It first proves a vertex-decomposition formula for polytopes and then extends to general convex bodies via a cone-volume to shadow lemma, yielding a practical integral (and Crofton-type) representation. By showing how Euclidean, Minkowski, Hilbert, and hyperbolic geometries arise as special or limiting cases, the paper unifies several classical surface-area formulas within a single shadow-averaging framework and provides computational primitives for non-Euclidean geometric tomography. The results facilitate Monte Carlo estimation and offer a pathway to efficient, locality-driven algorithms in Funk/Hilbert-type domains.

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 $\mathbb{R}^d$, under the Holmes-Thompson measure. Our formula is simple and is based on central projections to points on the boundary of $K$. We show that when $K$ is a convex polytope, the formula reduces to a weighted sum involving central projections at 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 single framework that unifies these classical surface area formulas.

Cauchy's Surface Area Formula in the Funk Geometry

TL;DR

This work develops explicit Cauchy-type formulas for Holmes-Thompson surface area in the Funk geometry induced by a convex body K, expressing area as an average of central shadows over all directions. It first proves a vertex-decomposition formula for polytopes and then extends to general convex bodies via a cone-volume to shadow lemma, yielding a practical integral (and Crofton-type) representation. By showing how Euclidean, Minkowski, Hilbert, and hyperbolic geometries arise as special or limiting cases, the paper unifies several classical surface-area formulas within a single shadow-averaging framework and provides computational primitives for non-Euclidean geometric tomography. The results facilitate Monte Carlo estimation and offer a pathway to efficient, locality-driven algorithms in Funk/Hilbert-type domains.

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 in , under the Holmes-Thompson measure. Our formula is simple and is based on central projections to points on the boundary of . We show that when is a convex polytope, the formula reduces to a weighted sum involving central projections at the vertices of . 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 single framework that unifies these classical surface area formulas.
Paper Structure (12 sections, 20 theorems, 88 equations, 13 figures)

This paper contains 12 sections, 20 theorems, 88 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. The Funk and Hilbert Geometries
  5. Holmes-Thompson Volume and Surface Area
  6. Cones
  7. Vertex Decomposition for Convex Polytopes
  8. Cauchy-Type Formulas for General Convex Bodies
  9. Connections to Other Geometries
  10. Hilbert and Hyperbolic Geometries
  11. Minkowski Geometry
  12. Justification of the Cone Volume Definition

Key Result

Theorem 1

Let $G$ and $K$ be two convex bodies in $\mathbb{R}^d$ such that $G \subset \mathop{\mathrm{int}}\nolimits(K)$. Then where $\sigma$ denotes the rotation-invariant surface measure on $S^{d-1}$.

Figures (13)

  • Figure 1: (a) The boundary point $v_K(u)$, (b) the central shadow $S_K(G,u)$ (translated so that $v_K(u)$ coincides with the origin), and (c) vertex decomposition.
  • Figure 2: (a) The difference body of a convex body and (b) the polar of a convex body.
  • Figure 3: (a) The Funk distance and (b) the Hilbert distance.
  • Figure 4: (a) The Finsler structure and (b) the Finsler ball for Hilbert (recentered about $x$).
  • Figure 5: (a) The cone $\mathop{\mathrm{cone}}\nolimits(G)$, (b) the normal cone $K_v^{\circ}$, and (c) $F_K(x)$.
  • ...and 8 more figures

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2: Vertex Decomposition for Polytopes
  • Lemma 2.1: Projection-Section Duality
  • Lemma 2.2
  • Lemma 2.3: Faifman Fai24
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • Proposition 2.6: Gnomonic Conversion
  • Lemma 3.1
  • ...and 26 more