Integral-Geometric Formulas for Perimeter in S^2, H^2, and Hilbert Planes

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dc.creatorAlexander, Ralph
dc.creatorBerg, I. D.
dc.creatorFoote, Robert L.
dc.date2005-03-15
dc.date.accessioned2026-07-07T05:17:59Z
dc.date.available2026-07-07T05:17:59Z
dc.descriptionWe develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central angle from an origin and the distance to the corresponding support line. The Minkowski formula for perimeter in E^2 involves polar coordinates and the geodesic curvature of the boundary of K. We generalize this to S^2 and H^2. In E^2 the Cauchy and Minkowski formulas are locally equivalent in the sense that the integrands are pointwise equal. In contrast, their generalizations in H^2 and S^2 are not locally equivalent.
dc.description26 Pages, 12 Figures. To appear in the Rocky Mountain Journal of Mathematics
dc.identifierhttps://arxiv.org/abs/math/0503313
dc.identifierhttp://arxiv.org/abs/math/0503313
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74507
dc.subjectDifferential Geometry
dc.subjectMetric Geometry
dc.subject53C65 (Primary), 52A38, 52A10, 26B15 (Secondary)
dc.titleIntegral-Geometric Formulas for Perimeter in S^2, H^2, and Hilbert Planes
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