Let two points x and y be picked randomly from a unit n-dimensional hypercube. The expected distance between the points Δ(n), i.e., the mean line segment length, is then Δ(n) = integral_0^1 ... integral_0^1_︸_(2n) sqrt((x_1 - y_1)^2 + (x_2 - y_2)^2 + ... + (x_n - y_n)^2)d x_1 ...d x_n d y_1 ...d y_n. This multiple integral has been evaluated analytically only for small values of n. The case Δ(1) corresponds to the line line picking between two random points in the interval [0, 1]. The first few values for Δ(n) are given in the following table.

cube line picking | hypercube point picking | mean line segment length | Robbins constant | square line picking | square point picking | square triangle picking | unit square integral