Cube Line Picking

The average distance between two points chosen at random inside a unit cube (the  case of hypercube line picking), sometimes known as the Robbins constant, is

			(1)
			(2)
			(3)

(OEIS A073012; Robbins 1978, Le Lionnais 1983).

The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form

(4)

The first even raw moments  for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).

Pick  points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table.

5	1.1180339887498
6	1.0606601482100
7	1
8	1
9	0.86602540378463
10	0.74999998333331
11	0.70961617562351
12	0.70710678118660
13	0.70710678118660
14	0.70710678118660
15	0.625

REFERENCES:

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.

Bolis, T. S. Solution to Problem E2629. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 277-278, 1978.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.

Finch, S. R. "Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 479-484, 2003.

Ghosh, B. "Random Distances within a Rectangle and between Two Rectangles." Bull. Calcutta Math. Soc. 43, 17-24, 1951.

Holshouser, A. L.; King, L. R.; and Klein, B. G. Solution to Problem E3217, "Minimum Average Distance between Points in a Rectangle." Amer. Math. Monthly 96, 64-65, 1989.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983.

Mathai, A. M.; Moschopoulos, P.; and Pederzoli, G. "Distance between Random Points in a Cube." J. Statistica 59, 61-81, 1999.

Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978.

Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.

Schroeppel, R. (results due to R. H. Hardin and N. J. A. Sloane) "points in a cube." math-fun@cs.arizona.edu posting, May 30, 1996.

Sloane, N. J. A. Sequences A073012, A160693, and A160694 in "The On-Line Encyclopedia of Integer Sequences."