quasi
Posts:
11,467
Registered:
7/15/05


Re: A tensided polygon has the following property ...
Posted:
Jun 11, 2014 4:29 AM


quasi wrote: >quasi wrote: >>quasi wrote: >>> >>>For each integer n >= 3, let f(n) be the greatest integer m >>>such that there exists an mgon in the plane whose boundary >>>can be covered by n lines. > >Clarification: > >The mgons 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 nondecreasing. >>> >>>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.
Ok, I see now that conjecture (2) is, in fact, obvious. After all, since each vertex is on two edges, each vertex must lie on two distinct covering lines, thus the vertice are at the intersection points of pairs of covering lines. But n distinct lines have at most C(n,2) intersection points, hence f(n) <= C(n,2).
>>(3) As n > oo, f(n)/C(n,2) > 1.
quasi

