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.

中文翻译：

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积分几何在 3D-Heisenberg 群中集合几何性质的应用

摘要 通过研究3D-Heisenberg 群H1 中的刚性运动群PSH(1)，我们定义了水平线集中的密度和测度。我们证明凸域 D ⊂ H1 的体积等于与 D 相交的所有水平线的弦长的积分。与经典积分几何一样，我们还定义了 PSH(1) 的运动密度，并表明与凸域 D ⊂ H1 相交的所有长度为 l 的段的度量可以由边界 ∂D 的 p 面积、D 的体积和 2l 表示。这两个结果都显示了[10]中使用变分方法导出的几何概率与自然几何量之间的关系。线段包含在凸域中的概率是作为我们结果的应用而获得的。