Hypercube Line Picking

Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then

(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)

where

			(9)
			(10)

(Bailey et al. 2006).

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

			(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). Attempting to reduce to quadratures gives closed-form pieces with the exception of the single piece

			(17)
			(18)
			(19)

which appears to be difficult to integrate in closed form (Bailey et al. 2007, p. 272).

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