quasi wrote: >quasi wrote: >> >>For each integer n >= 3, let f(n) be the greatest integer m >>such that there exists an m-gon in the plane whose boundary >>can be covered by n lines.
The m-gons are required to be piecewise linear simple closed curves with no pair of adjacent edges lying on the same line.
>>Some instant observations: >> >>(1) f(n) >= n, for all n. >> >>(2) f is non-decreasing. >> >>An easy observation: >> >>(3) f(n) >= 2n, for all n >= 5. >> >>Some tentative data: >> >> f(3)=3 >> f(4)=4 >> f(5)=10 >> f(6)=12 >> f(7)=18 >> f(8)=21 >> f(9)=32 >> f(10)=36 >> f(11)=48 >> >>Corrections and/or extensions to the above are welcome. > >A few conjectures: > >(1) f is strictly increasing. > >(2) f(n) <= C(n,2), for all n. > >(3) As n -> oo, f(n)/C(n,2) -> 1.