Integral Geometry of Pairs of Lines and Planes

In this paper, we present some results obtained previously in Cufí et al. (J. Math. Anal. Appl. 458(1):436–451 (2018); Mathematika 65(4):874–896 (2019); Rend. Circ. Mat. Palermo (2) 69(3):1115–1130 (2020); Arch. Math. (Basel) 117(5):579–591 (2021)) related to convex sets in the plane and in the space. In the plane, we deal with Hurwitz’s inequality, which provides an upper bound of the isoperimetric deficit of a convex set K in terms of the area of the evolute of the boundary of K. We improve this inequality finding strictly positive lower bounds for the Hurwitz’s deficit, these bounds involving the visual angle of the boundary of K. In a different look, we provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz, and Masotti. Also, we interpret these formulas from the point of view of Integral Geometry of pairs of lines. In the space, we deal with integrals of invariant measures of pairs of planes, expressing some of these integrals in terms of functions of the visual dihedral angle of the convex set. As a consequence of our results, we evaluate the deficit in a Crofton-type inequality due to Blaschke.