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Jung's Theorem


Jung's theorem states that the generalized diameter D of a compact set X in R^n satisfies

 D>=Rsqrt((2(n+1))/n),

where R is the circumradius of X (Danzer et al. 1963).

This gives the special case that every finite set of points in two dimensions with geometric span d has an enclosing circle with radius no greater than sqrt(3)d/3 (Rademacher and Toeplitz 1957, p. 104).


See also

Enclosing Circle, Geometric Span, Minimal Enclosing Circle

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References

Danzer, L.; Grünbaum, B.; and Klee, V. "Helly's Theorem and its Relatives." In Convexity: Proceedings, Symposia in Pure Mathematics. Providence RI: Amer. Math. Soc., pp. 101-180, 1963.Jung, H. W. E. "Über die kleinste Kugel, die eine räumliche Figur einschliesst." J. reine angew. Math. 123, 241-257, 1901.Jung, H. W. E. "Über den kleinsten Kreis, der eine ebene Figur einschliesst." J. reine angew. Math. 137, 310-313, 1910.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983.Rademacher, H. and Toeplitz, O. "The Spanning Circle of a Finite Set of Points." §16 in The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 103-110, 1957.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 128, 1991.

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Jung's Theorem

Cite this as:

Weisstein, Eric W. "Jung's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JungsTheorem.html

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