Abstract

In the present paper a formula for calculation of the density function \(f_{\rho}(x)\) of the distance between two independent points randomly and uniformly chosen in a bounded convex body \(D\) is given. The formula permits to find an explicit form of density function \(f_{\rho}(x)\) for body \(D\) with known chord length distributions. In particular, we obtain an explicit expression for \(f_{\rho}(x)\) in the case of a ball of diameter \(d\) in \(R^{3}\).

中文翻译：

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人体中两个随机点之间的距离分布，从$$ \ boldsymbol {R} ^ {\ mathbf {3}} $$

摘要

在本文中，给出了计算有界凸体\（D \）中随机且均匀选择的两个独立点之间距离的密度函数\（f _ {\ rho}（x）\）的 公式。该公式允许找到具有已知弦长分布的物体\（D \）的密度函数\（f _ {\ rho}（x）\）的显式形式。特别是，如果在（R ^ {3} \）中直径为\（d \）的球的情况下，我们得到\（f _ {\ rho}（x）\）的显式表达式 。