Sunday, August 7, 2016

Points In A Circle Problem: A Wild Guess for N = 3

With reference to the optimization problems in my last post, I found myself most interested by the N = 3 case:
Place 3 points in the interior of a unit disk so that the largest triangle not containing any of the points has least possible area.
I should actually clean up my language here, because the maximum area could be attained by two or more different triangles. It ought to say, "a largest triangle," not "the largest triangle." Or perhaps I could say:
Place 3 points in the interior of a unit disk so that the largest area among all triangles not containing any of the points is as small as possible.

In today's post, I thought I'd share my first wild guess at a solution to the N = 3 problem:



The dots are each located about 0.321 units from the center of the disk, and the configuration has 120-degree rotational symmetry.

Given my opening move, what is player two's best response? What is the largest area of any triangle that doesn't contain any of my points within its interior? Feel free to email me a sketch at zimblogzimblog@gmail.com.

Remember, it's OK if the boundary of your triangle touches one or more of the points, and it's OK if one or more of your triangle's vertices lies on the boundary of the disk.

So far, the greatest area I've been able to enclose with a triangle that doesn't contain the above three points is about 0.829 square units. But I haven't searched systematically or at length, so I'm far from certain that this is the greatest area possible. If any reader sketch appears to exceed that value, I'll let you know.

One more reflection on the problem: is it well posed? There are infinitely many triangles not containing the three given points, so one must consider the possibility that among them no greatest area exists. I guess I doubt that happens here—basically, because we allow touching of boundaries—but I haven't proven it.

No comments: