2
$\begingroup$

I understand that there are sets of 7 points on a circle that can be fully shattered using triangles.But, it is not clear to me why it cannot shatter 8 points.

Is there any intuitive way of arriving at the conclusion that it can't shatter 8 points? Is there a simple explanation without using much advanced geometry tools?

$\endgroup$
  • 1
    $\begingroup$ The tags seem unmotivated by anything in the question. statistics? classification? machine-learning? $\endgroup$ – Gerry Myerson Oct 4 '18 at 23:54
  • $\begingroup$ Sorry for my ignorance... The answer below gives me a guess of what "shattering" may mean, but then I don't see how a triangle can "shatter" 7 pts on a circle... Can you please define shattering, or give a 7 pt example? Thx. $\endgroup$ – Yaakov Baruch Oct 5 '18 at 10:08
  • 1
    $\begingroup$ 7 point example: if you have 7 points along the perimeter of a circle and say they are randomly labelled + and -. Using a triangle I will be able to separate them into positive and negative classes. This is because, with 7 points, you can have max of 3 contiguous blocks of -. (Note that they are arranged in circular way). Now, you can separate the three blocks using 3 corners of your triangle. Hence 7 points can be shattered. $\endgroup$ – Jyotish Robin Oct 5 '18 at 16:03
  • $\begingroup$ @JyotishRobin. Thank you. That also clarifies what shattering means. $\endgroup$ – Yaakov Baruch Oct 6 '18 at 19:37
4
$\begingroup$

Let $x_1, \dots, x_8$ be $8$ points on a circle in this order. Then there is no triangle containing exactly $x_2, x_4, x_6, x_8$ in its interior and the other four points in its exterior. Intuitively, it would have to intersect the circle in at least $8$ points, but any triangle intersects a circle in at most $6$ points.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.