The kinematic formula in the 3D-Heisenberg group

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.

15 pages, 3 figures

References in corpus (2)

• Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n
• An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

Cited by in corpus (3)

• An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group
• Tubular neighborhoods in the sub-Riemannian Heisenberg groups
• Cauchy surface area formula in the Heisenberg groups