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

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.