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

作者：Huang, Yen-Chang^*

来源：Analysis and Geometry in Metric Spaces, 2016, 4(1): 425-435.

DOI：10.1515/agms-2016-0020

• Heisenberg Group
• kinematic formula
• Integral Geometry
• CR-manifolds
• method of moving frames

By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H-1, we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D subset of H-1 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 subset of H-1 can be represented by the p- area of the boundary sigma 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.

• 出版日期2016-1
• 单位厦门大学