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Cauchy’s Surface Area Formula in the Funk Geometry

Authors Sunil Arya , David M. Mount

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LIPIcs.SoCG.2026.8.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Convexity
  • Cauchy’s formula
  • Funk geometry
  • Hilbert geometry
  • Crofton’s formula
  • Holmes-Thompson surface area

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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 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.

Cite As Get BibTex

Sunil Arya and David M. Mount. Cauchy’s Surface Area Formula in the Funk Geometry. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.8

Author Details

Sunil Arya
  • Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong
David M. Mount
  • Department of Computer Science, University of Maryland, College Park, MD, USA

Funding

  • Arya, Sunil: Research Grants Council of Hong Kong, China project number 16214721.

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