Cauchy's Surface Area Formula in the Funk Geometry

TL;DR

This paper extends Cauchy's surface area formula to the Funk geometry, providing a unified framework for classical surface area formulas across various geometries.

Contribution

It establishes an analog of Cauchy's formula for Funk geometry and generalizes Crofton's formula, unifying several classical surface area formulas.

Findings

Derived a Funk geometry version of Cauchy's surface area formula.

Reduced the formula to vertex contributions for convex polytopes.

Unified classical surface area formulas within the Funk geometric framework.

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.