Integral-geometric formulas for perimeter in S2, H2 and Hilbert planes

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. In the projective model of H2 we have P = (1/2) integral omega d phi. Here omega is the Euclidean length of the projection of K from the ideal boundary point R = (cos phi, sin phi) onto the diametric line perpendicular to the radial line to R (the image of K may contain points outside the model). We show that the standard Cauchy formula P = integral sinh r d omega in H-2 follows, where w is a central angle perpendicular to a support line and r is the distance to the support line. The Minkowski formula P = integral kg rho(2) d theta in E-2 generalizes to P = 1/(4 pi(2)) integral k(g)L(rho)(2) d theta + k/2 pi integral A(rho) ds in H-2 and S-2. Here (rho, theta) and kappa(g) are, respectively, the polar coordinates and geodesic curvature of partial derivative K, kappa is the (constant) curvature of the plane, and L(p) and A(p) are, respectively, the perimeter and area of the disk of radius p. In E2 this is locally equivalent to the Cauchy formula P = integral r d omega in the sense that the integrands are pointwise equal. In contrast, the corresponding Minkowski and Cauchy formulas are not locally equivalent in H-2 and S-2.