# Minimize total area bounded by $N$ lines in general position

Suppose we have $$N$$ lines in general position (any two lines, but no three lines, meet at a point) ($$N\geq 3$$). Let the smallest bounded region have area $$1$$. Determine the minimum (or possibly infimum) of the total bounded area.

For example, $$3$$ lines create one triangular region while $$4$$ create one quadrangular region and two triangular regions (the complete region thus bounded is called a complete quadrilateral). I know that $$N$$ lines in general position will create $${N+1\choose 2}+1$$ regions, $${N+1\choose 2}+1-2N=\frac{(N-1)(N-2)}{2}$$ of those bounded on all sides by line segments, but I wanted to figure out how to make the regions maximally “equal” and how close it’s possible to get to the ideal minimum of $$\frac{(N-1)(N-2)}{2}$$ for a given $$N$$ (where all bounded regions have area $$1$$). I have little to no experience in the field of maximizing or minimizing geometrical quantities (especially in such a large mathematical space in which to minimize/maximize as this problem gives), so solving this is entirely beyond my experience.

It is easy to show that there is no maximal bounded area for $$N\geq 4$$ by drawing some three lines arbitrarily close to meeting at a point. In this case, the resulting triangular region becomes arbitrarily close to a point while maintaining an area of $$1$$ in the problem's scaling, and the area of the remaining regions grows arbitrarily large by comparison. But I really don't know how to do any of this when it comes to minimizing the total bounded area.

For $$N=3$$, with only one region, the minimum area is trivially $$1$$. For $$N=4$$, some fiddling around on Desmos has convinced me that the minimum area is indeed the ideal of $$3$$. For $$N=5$$, I believe that the minimum area still remains the ideal of $$6$$ (although here my fiddling around on Desmos gets much more nonrigorous and guesswork-y). For $$N\geq 6$$, however, any proofs or even good guesses of any sort elude me entirely.

If finding the exact minimum area (or infimum in case there’s somehow a minimum that can be approached but not actually reached, which I would not expect) is in fact too difficult to do well, I would also be interested in the growth rate of the actual minimal area relative to the ideal as $$N\to\infty$$. Thank you for your help! :)

• Can the angles in a quadrilateral ( whose four lengths are given) be always changed to make it cyclic... of maximum area? Jul 14, 2022 at 22:59
• I thought I understood the problem, but I lost you soon, when you talk about $N=4$. If I build a square, where do the 2 triangles come from? It's a peculiar configurations, but how can you exclude such a configuration from the optimization process? What is the "not enough general configuration" you're using? Or am I missing something? Aug 30, 2022 at 21:00
• @basics It's in general position. Any two lines, but no three lines, meet at a point. So no lines parallel to any others, and no three meeting at a point. So a square is impossible. Aug 30, 2022 at 22:21
• ok, I didn't understand that any pair of lines MUST meet. I'll start thinking about the problem in the next days Aug 30, 2022 at 22:27
• Well, yeah- I did say "any two lines...meet at a point". But thank you. Aug 30, 2022 at 22:28