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

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

2005-03-15

arxiv.org/abs/math/0503313v1

Mathematics

Differential Geometry

26 Pages, 12 Figures. To appear in the Rocky Mountain Journal of Mathematics

Scientific paper

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.

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