Magic Geometric Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $E$ be a compact connected subset of $d$-dimensional Euclidean Space. Gross (1964) and Stadje (1981) proved that there is a unique Real Number $a(E)$ such that for all $x_1$, $x_2$, ..., $x_n \in E$, there exists $y\in E$ with

$${1\over n}\sum_{j=1}^n \sqrt{\sum_{k=1}^d (x_{j,k}-y_k)^2} = a(E).$$ (1)

The magic constant $m(E)$ of $E$ is defined by

$$m(E)={a(E)\over{\rm diam}(E)},$$ (2)

where

$${\rm diam}(E)\equiv \max_{u,v\in E}\sqrt{\sum_{k=1}^d (u_k-v_k)^2}\,.$$ (3)

These numbers are also called Dispersion Numbers and Rendezvous Values. For any $E$, Gross (1964) and Stadje (1981) proved that

$${\textstyle{1\over 2}}\leq m(E)<1.$$ (4)

If $I$ is a subinterval of the Line and $D$ is a circular Disk in the Plane, then

$$m(I)=m(D)={\textstyle{1\over 2}}.$$ (5)

If $C$ is a Circle, then

$$m(C)={2\over\pi}=0.6366\ldots.$$ (6)

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 $T$

$$0.6675276\leq m(T)\leq 0.6675284.$$ (7)

Denote the Maximum value of $m(E)$ in $n$-D space by $M(n)$. Then

$M(1)$	${\textstyle{1\over 2}}$
$M(2)$	$m(T)\leq M(2)\leq {2+\sqrt{3}\over 3\sqrt{3}}<0.7182336$
$M(d)$	${d\over d+1}\leq M(d)\leq {[\Gamma({\textstyle{1\over 2}}d)]^2 2^{d-2}\sqrt{2d}\over \Gamma(d-{\textstyle{1\over 2}})\sqrt{(d+1)\pi}} < \sqrt{d\over d+1}$

where $\Gamma(z)$ 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. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/magic/magic.html

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.

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