I know that when $n$ lines are drawn in the plane, the maximum possible number of regions the plane is divided into is $\frac{n(n+1)}2+1$. This maximum is achieved when no two of the lines are parallel, and no three are collinear.

When the plane is divided into regions by $n$ lines, some of these regions will be bounded, and some will be infinite. What is the maximum possible number of infinite regions in such an arrangement?

  • $\begingroup$ What you mean is that the number of regions into which $n$ lines, no three of which intersect at the same point, divide a plane is $\frac{n(n + 1)}{2} + 1$. $\endgroup$ – N. F. Taussig Oct 1 '19 at 9:56
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    $\begingroup$ I am confused of "infinite areas". Can you explain more? $\endgroup$ – Isaac YIU Math Studio Oct 1 '19 at 9:58
  • $\begingroup$ Wrong. n parallel lines divide a plain into n + 1 areas. $\endgroup$ – William Elliot Oct 1 '19 at 10:45
  • $\begingroup$ You have to impose a condition such as 'no 3 lines are concurrent' or something like that. Else the number of lines depends on the configuration. $\endgroup$ – Sam Oct 1 '19 at 11:58
  • $\begingroup$ Are you asking how many of the regions are unbounded? $\endgroup$ – saulspatz Oct 1 '19 at 12:10

Imagine taking a circle $C$ which is large enough such that (i) all of the intersection points between lines in the arrangement lie inside $C$ and (ii) every line intersects $C$ twice.

It follows from (i) that bounded regions do not intersect $C$ (since all their vertices are inside $C$). Furthermore, any unbounded region must intersect $C$ (The region must have points outside $C$ since it's unbounded. Take a point on the boundary of the region lying outside $C$, follow the boundary line towards $C$ -- it can't intersect another boundary before hitting $C$ by (i))

Now imagine making a single circuit travelling around $C$. There will be $2n$ distinct points at which $C$ intersects one of the lines (since each line intersects twice and (i) implies that no two lines intersect $C$ in the same place). So we only change regions $2n$ times before returning back where we started, implying we've passed through at most $2n$ regions total. So the number of unbounded regions is at most $2n$.

Equality holds so long as we don't encounter the same region twice as we travel around the circle (which I believe only happens if we have $n$ parallel lines). For example, we get equality if we just take $n$ concurrent lines. The number of regions is $2n$ (very far from maximal), but they're all unbounded.


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