Distribution of the Distance Between Two Random Points in a Body from $$\boldsymbol{R}^{\boldsymbol{n}}$$

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 with known chord length distributions. In particular, we obtain an explicit expression for $$f_{\rho}(x)$$ in the case of a ball of diameter $$d$$ . A simulation model is suggested to calculate empirically the cumulative distribution function of the distance between two points in a body from $$R^{n}$$ , where explicit form of the function is hard to obtain. In particular, simulation is performed for balls and ellipsoids in $$R^{n}$$ .