Cauchy's Surface Area Formula in the Funk Geometry

2026-01-23

AI Summary

This work extends the classical Cauchy surface area formula from Euclidean geometry to the setting of Funk geometry induced by convex bodies. By employing the Holmes–Thompson measure and central projections from boundary points, the authors establish a Funk surface area formula applicable to general convex bodies and convex polytopes, along with a corresponding Crofton-type formula. This study presents the first integral-geometric relations of this kind in Funk geometry, unifying and generalizing classical results from Euclidean, Minkowski, Hilbert, and hyperbolic geometries. The framework provides a cohesive theoretical foundation for geometric measure theory in non-Euclidean spaces, highlighting the intrinsic connections among these diverse geometric structures through the lens of convexity and projective invariance.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Cauchy's surface area formula
• Funk geometry
• non-Euclidean geometry
• convex body
• surface area

Innovation

• Funk geometry
• Cauchy's surface area formula
• Holmes-Thompson measure
• central projection
• Crofton's formula