Magic Geometric Constants

Let  be a compact connected subset of -dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number  such that for all , , ..., , there exists  with

(1)

The magic constant  of  is defined by

(2)

where

(3)

These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that

(4)

If  is a subinterval of the line and  is a circular disk in the plane, then

(5)

If  is a circle, then

(6)

(OEIS A060294). An expression for the magic constant of an ellipse in terms of its semimajor and semiminor axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux triangle 

(7)

Denote the maximum value of  in -dimensional space by . Then

where  is the gamma function (Nikolas and Yost 1988).

An unrelated quantity characteristic of a given magic square is also known as a magic constant.

REFERENCES:

Finch, S. R. "Rendezvous Constants." §8.21 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 537-542, 2003.

Cleary, J.; Morris, S. A.; and Yost, D. "Numerical Geometry--Numbers for Shapes." Amer. Math. Monthly 95, 260-275, 1986.

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994.

Gross, O. The Rendezvous Value of Metric Space. Princeton, NJ: Princeton University Press, pp. 49-53, 1964.

Nikolas, P. and Yost, D. "The Average Distance Property for Subsets of Euclidean Space." Arch. Math. (Basel) 50, 380-384, 1988.

Sloane, N. J. A. Sequence A060294 in "The On-Line Encyclopedia of Integer Sequences."

Stadje, W. "A Property of Compact Connected Spaces." Arch. Math. (Basel) 36, 275-280, 1981.