Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group

Abstract

AbstractBy studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments with length l intersecting a convex domain D ⊂ H1 can be represented by the p-area of the boundary ∂D, the volume of D, and 2l. Both results show the relationship between geometric probability and the natural geometric quantity in [10] derived by using variational methods. The probability that a line segment be contained in a convex domain is obtained as an application of our results.