Jung's Theorem in the Plane

Theorem

Let $S \subseteq \R^2$ be a compact region in a Euclidean plane.

Let $d$ be the diameter of $S$.


Then there exists a circle $C$ with radius $r$ such that:

$r = d \dfrac {\sqrt 3} 3$

such that $S \subseteq C$.


The parameter $\dfrac {\sqrt 3} 3$ can also be presented as $\dfrac 1 {\sqrt 3}$, and evaluates approximately as:

$\dfrac {\sqrt 3} 3 \approx 0 \cdotp 57735 \, 02691 \ldots$

This sequence is A020760 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

This is an instance of Jung's Theorem, setting $n = 2$.

$\blacksquare$


Source of Name

This entry was named for Heinrich Wilhelm Ewald Jung.


Sources

  • 1910: Heinrich Jung: Über den kleinsten Kreis, der eine ebene Figur einschließt (J. reine angew. Math. Vol. 137: pp. 310 – 313)
  • 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,57735 02691 \ldots$
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