Hypercube Line Picking

Hypercube line picking considers two points and chosen independently and uniformly from a unit -dimensional hypercube. Their expected distance , i.e., the mean line segment length, is

(1)

This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .

The first few values for are given in the following table.

	OEIS
1	--	0.3333333333...
2	A091505	0.5214054331...
3	A073012	0.6617071822...
4	A103983	0.7776656535...
5	A103984	0.8785309152...
6	A103985	0.9689420830...
7	A103986	1.0515838734...
8	A103987	1.1281653402...

The function satisfies

(2)

(Anderssen et al. 1976), plotted above together with the actual values.

M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that

(3)

The first few values are

			(4)
			(5)
			(6)
			(7)

In the limit as , these have values for , 2, ... given by times 2/3, 6/5, 50/21, 38/9, 74/11, ... (OEIS A103990 and A103991).

This is equivalent to computing the box integral

(8)

for , where

			(9)
			(10)

(Bailey et al. 2007).

These give closed-form results for , 2, 3, 4, and 5:

			(11)
			(12)
			(13)
			(14)
			(15)

where is a Clausen function, is Catalan's constant, and

(16)

The case above was published for the first time in this work; the simplified form given above is due to Bailey et al. (2007). The expression for uses the auxiliary integral

			(17)
			(18)
			(19)

A closed form for in terms of Clausen functions was obtained by Bailey et al. (2010, eq. 61).

The value obtained for cube line picking is sometimes known as the Robbins constant.