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.

翻译：柯西表面积公式将凸体的表面积表示为所有方向上正交投影的平均面积。尽管该工具在欧氏几何中具有基础性地位（应用范围从几何层析成像到逼近理论），但在非欧几里得框架中的推广仍鲜有探索。本文针对由凸体$K \subset \mathbb{R}^d$诱导的Funk几何，建立了其Holmes--Thompson表面积的柯西公式的类比形式。该公式基于中心投影到$K$边界点的结构。我们证明：当$K$为凸多面体时，该公式可简化为与$K$顶点相关的加权求和。最后，作为分析的推论，我们推导出Funk几何中表面积的克罗夫顿公式的推广形式。通过将欧氏、闵可夫斯基、希尔伯特及双曲几何视为Funk框架的极限或特例，我们的结果为这些经典表面积公式提供了统一的理论框架。