1. Integral-Geometric Formulas for Perimeter in S^2, H^2, and Hilbert Planes
- Author
-
Alexander, Ralph, Berg, I. D., and Foote, Robert L.
- Subjects
Mathematics - Differential Geometry ,Mathematics - Metric Geometry ,53C65 (Primary), 52A38, 52A10, 26B15 (Secondary) - Abstract
We 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., Comment: 26 Pages, 12 Figures. To appear in the Rocky Mountain Journal of Mathematics
- Published
- 2005