Points contained in a disk I have a question, but not sure how to prove this.
We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points.
Conjecture: There must exist three of these points which are not contained in a disk of radius $a$.
Any idea about how to prove this?
Thanks.
 A: Start by choosing two of the points $A$ and $B$ such that all other points lie on the same side of the line $AB$. (This can be done by moving a line from the distance to the points until it meets a point, and then turning the line until you meet another point.) It there are other points on the line AB, you should choose $A$ and $B$ such that the other points lie between $A$ and $B$.
Now consider circles through these two points with various radius where the larger part of the circle lies on the side of the other points.
Then a circle with larger radius covers all the area of a circle of smaller radius (inside the half space where the points lie). Hence, the biggest circle you need to cover $A$, $B$ and a third point already covers all points.
So, $A$, $B$ and the third point for which one needed this biggest circle are the three points in your conjecture.
A: There is a classical result of convex geometry, called Helly's theorem (http://en.wikipedia.org/wiki/Helly%27s_theorem). It states that if you have $n$ convex subsets of $R^d$ and any $d+1$ of these convex subsets have nontrivial intersection, then all of them have nontrivial intersection.
For your question you just apply this theorem to the balls of radius $a$ around your points.
