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.

翻译：柯西表面积公式将凸体的表面积表示为其在所有方向上正交投影面积的平均值。尽管该工具在欧几里得几何中具有基础性地位，其应用范围涵盖几何层析成像到逼近理论等多个领域，但在非欧几里得设定下的推广仍较少被探索。本文针对由$\mathbb{R}^d$中凸体$K$诱导的Funk几何，在Holmes-Thompson测度下建立了柯西公式的一个类比形式。我们的公式形式简洁，其基础在于向$K$边界点的中心投影。我们证明当$K$为凸多面体时，该公式可简化为涉及$K$顶点处中心投影的加权和。最后，作为分析的一个推论，我们导出了Funk几何中表面积的克罗夫顿公式的推广。通过将欧几里得、闵可夫斯基、希尔伯特及双曲几何视为Funk设定的极限或特例，我们的结果提供了一个统一这些经典表面积公式的单一框架。