Distribution of the Distance between Two Random Points in a Body from $$\boldsymbol{R}^{\mathbf{3}}$$

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 $$R^{3}$$ .