In a given set of points, Prove that the two farthest points are the vertices of the convex hull,

How can i get the accurate proof, so that the question can be explained in the class

  • 4
    $\begingroup$ Can you state the definition of the convex hull (and maybe of a convex set as well) and tell us what you tried so far? $\endgroup$
    – usul
    Sep 19 '16 at 17:28
  • 4
    $\begingroup$ What did you try? Where did you get stuck? We're happy to help with conceptual questions but just solving exercise-style problems for you is unlikely to really help. $\endgroup$ Sep 19 '16 at 18:49
  • $\begingroup$ Hint: If two points are not vertices of the convex hull, how can you find points that are further away? $\endgroup$
    – gnasher729
    Sep 19 '16 at 21:22
  • $\begingroup$ cs.stackexchange.com/questions/23646/… - and it's NOT closed $\endgroup$
    – HEKTO
    Jan 26 '17 at 16:43

I'm assuming you asked about points on two-dimensional plane.

Let $P_1$ and $P_2$ are these two points, such that the (Euclidean) distance $D$ between them is maximal over all the pairs of points in the given point set. All the other points must be inside an intersection of two disks with radius $D$ - one disk with center in the $P_1$ and another disk with center in the $P_2$.

So, we have a curvilinear rhombus with diagonal $(P_1, P_2)$ and circular sides with radius $D$, which must contain both all points of the given set and also their convex hull.


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