Given a set of points of diameter in dimensional Euclidean space it is trivial to see that it can be covered by a ball of radius . But the following theorem by Jung improves the result by a factor of about , and is the best possible.
Theorem 1 [ Jung’s Theorem: ] Let be a set of points in of diameter . Then there is a ball of radius covering .
Proof: We first prove this theorem for sets of points with and then extend it to an arbitrary point set. If then the smallest ball enclosing exists. We assume that its center is the origin. Denote its radius by . Denote by the subset of points such that for . It is easy to see that is in fact non empty.
Observation: The origin must lie in the convex hull of . Assuming the contrary, there is a separating hyperplane such that lies on one side and the origin lies on the other side of (strictly). By assumption, every point in is distance strictly less than from the origin. Move the center of the ball slightly from the origin, in a direction perpendicular to the hyperplane towards such that the distances from the origin to every point in remains less than . However, now the distance to every point of is decreased and so we will have a ball of radius strictly less than enclosing which is a contradiction to the minimality of .
Let where and because the origin is in the convex hull of so we have nonnegative such that,
Fix a . Then we have,
Adding up the above inequalities for all values of , we get
Thus we get since and the function is monotonic. So we have immediately .
The remainder of the proof uses the beautiful theorem of Helly. So assume is any set of points of diameter . With each point as center draw a ball of radius . Clearly any of these balls intersect. This is true because the center of the smallest ball enclosing of the points is at most away from each of those points. So we have a collection of compact convex sets, any of which have a nonempty intersection. By Helly’s theorem all of them have an intersection. Any point of this intersection can be chosen to be the center of a ball of radius that will enclose all of .