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)
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The magic constant of is defined by
(2)
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where
(3)
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These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that
(4)
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If is a subinterval of the line and is a circular disk in the plane, then
(5)
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If is a circle, then
(6)
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(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)
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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.