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.



Points contained in a disk
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.


