Integral-geometric Formulas for Perimeter in S, H 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 H2 we get a formula that appears to be new. In the projective model of H2 we have P = (1/2) ∫ w dφ. Here w is the Euclidean length of the projection of K from the ideal boundary point R = (cosφ, sinφ) 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 = ∫ sinh r dω in H2 follows, where ω is a central angle perpendicular to a support line and r is the distance to the support line. The Minkowski formula P = ∫ κgρ dθ in E2 generalizes to P = 1/(4π2) ∫ κgL(ρ) dθ+ k/2π ∫ A(ρ) ds in H2 and S2. Here (ρ, θ) and κg are, respectively, the polar coordinates and geodesic curvature of ∂K, k is the (constant) curvature of the plane, and L(ρ) and A(ρ) are, respectively, the perimeter and area of the disk of radius ρ. In E2 this is locally equivalent to the Cauchy formula P = ∫ r dω in the sense that the integrands are pointwise equal. In contrast, the corresponding Minkowski and Cauchy formulas are not locally equivalent in H2 and S2.