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

Yen-Chang Huang. "Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group." Analysis and Geometry in Metric Spaces 4.1 (2016): 425-435. <http://eudml.org/doc/288029>.

@article{Yen2016,
abstract = {By 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.},
author = {Yen-Chang Huang},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Heisenberg Group; kinematic formula; Integral Geometry; CR-manifolds; method of moving frames; Heisenberg group; integral geometry; method of moving frames},
language = {eng},
number = {1},
pages = {425-435},
title = {Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group},
url = {http://eudml.org/doc/288029},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Yen-Chang Huang
TI - Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group
JO - Analysis and Geometry in Metric Spaces
PY - 2016
VL - 4
IS - 1
SP - 425
EP - 435
AB - By 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.
LA - eng
KW - Heisenberg Group; kinematic formula; Integral Geometry; CR-manifolds; method of moving frames; Heisenberg group; integral geometry; method of moving frames
UR - http://eudml.org/doc/288029
ER -