Some integral geometric results in the simply isotropic space (Q1407624)

Let \(I_3^{(1)}\) denote the simply isotropic space defined as \(P_3({\mathbb R})\) in which the absolute consists of a plane \(\omega\) and two complex conjugate straight lines \(f_1\), \(f_2\) with a real intersection point in \(\omega\) [\textit{H. Sachs}, Isotrope Geometrie des Raumes. Braunschweig: Vieweg (1990; Zbl 0703.51001)]. Let \(B_6^{(1)}\) be the group of simply isotropic motions in \(I_3^{(1)}\). Several results concerning the measurability w.r.t. \(B_6^{(1)}\) in \(I_3^{(1)}\) of the sets of points, sets of straight lines, sets of pairs, triples and quadruples of points, sets of pairs and triples of planes are given. Some Crofton type formulas are also obtained.