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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 1:26 AM

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.
>
>(3) As n -> oo, f(n)/C(n,2) -> 1.

quasi

Date Subject Author
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