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Topic: A ten-sided polygon has the following property ...
Replies: 40   Last Post: Jun 12, 2014 12:44 AM

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 quasi Posts: 12,067 Registered: 7/15/05
Re: A ten-sided 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 m-gon in the plane whose boundary
>>>can be covered by n lines.

>
>Clarification:
>
>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.

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

Date Subject Author
6/8/14 Port563
6/8/14 James Waldby
6/9/14 Port563
6/9/14 William Elliot
6/9/14 Robin Chapman
6/9/14 Port563
6/9/14 Peter Percival
6/10/14 William Elliot
6/9/14 Peter Percival
6/9/14 Robin Chapman
6/9/14 Phil Carmody
6/9/14 Brian Q. Hutchings
6/10/14 Port563
6/10/14 Leon Aigret
6/10/14 Port563
6/10/14 Leon Aigret
6/10/14 Port563
6/10/14 Leon Aigret
6/10/14 Port563
6/10/14 Phil Carmody
6/10/14 Phil Carmody
6/10/14 Port563
6/10/14 Phil Carmody
6/10/14 Port563
6/10/14 thenewcalculus@gmail.com
6/10/14 Brian Q. Hutchings
6/10/14 Brian Q. Hutchings
6/11/14 Port563
6/10/14 Waldek Hebisch
6/11/14 quasi
6/11/14 quasi
6/11/14 quasi
6/11/14 quasi
6/12/14 quasi
6/12/14 quasi
6/11/14 Phil Carmody
6/11/14 quasi
6/11/14 Phil Carmody
6/11/14 Port563
6/11/14 Leon Aigret
6/11/14 quasi