Jung's Theorem in the Plane
Jump to navigation
Jump to search
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): $0,57735 02691 \ldots$