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.
