Jung's Theorem

Theorem

Let $S \subseteq \R^n$ be a compact subspace of an $n$-dimensional Euclidean space.

Let $d = \ds \max_{x, y \mathop \in S} \map d {x, y}$ be the diameter of $S$.

Then there exists a closed ball ${B_r}^-$ with radius $r$ such that:

$r = d \sqrt {\dfrac n {2 \paren {n + 1} } }$

such that $S \subseteq {B_r}^-$.

Jung's Theorem in the Plane

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 $S \subseteq C$.

Proof

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Source of Name

This entry was named for Heinrich Wilhelm Ewald Jung.

Sources

• 1901: Heinrich Jung: Über die kleinste Kugel, die eine räumliche Figur einschließt (J. reine angew. Math. Vol. 123: pp. 241 – 257)