Cauchy's Surface Area Formula in the Funk Geometry

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.