Magic geometric constants are normalized average-distance values of compact connected subsets of Euclidean space containing at least two points. Let
be such a 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)
|
The number 
is also called the rendezvous value, while 
is a dispersion number. 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. Nickolas
and Yost (1988) showed that for a Reuleaux triangle 
 |
(7)
|
Denote the maximum value of 
in 
-dimensional space by 
. Then
 |  |
 |  |
 | ![d/(d+1)<=M(d)<=([Gamma(1/2d)]^22^(d-2)sqrt(2d))/(Gamma(d-1/2)sqrt((d+1)pi))<sqrt(d/(d+1))](/images/equations/MagicGeometricConstants/Inline25.svg) |
where 
is the gamma function (Nickolas and Yost 1988).
The upper bound for 
is 0.678442037... (Pillichshammer 2001).
An unrelated quantity characteristic of a given magic square is also known as a magic constant.
