The kinematic formula in the 3D-Heisenberg group

TL;DR

This paper extends integral geometry to the 3D-Heisenberg group by defining measures for horizontal lines, establishing a kinematic formula, and linking geometric probability with natural geometric quantities.

Contribution

It introduces a new framework for measuring and analyzing the geometry of the 3D-Heisenberg group, including a kinematic formula and probabilistic interpretations.

Findings

Volume of convex domains equals the integral of chord lengths over intersecting horizontal lines.

Defined the density and measure for sets of horizontal lines in the 3D-Heisenberg group.

Established a relationship between geometric probability and natural geometric quantities.

Abstract

By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group $H_1$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain $D\subset H_1$ is equal to the integral of length of chord over all horizontal lines intersecting $D$. As the classical result in integral geometry, we also define the kinematic density for $PSH(1)$ and show the probability of randomly throwing a vector $v$ interesting the convex domain $D\subset D_0$ under the condition that $v$ is contained in $D_0$. Both results show the relationship connecting the geometric probability and the natural geometric quantity in Cheng-Hwang-Malchiodi-Yang's work approached by the variational method.