

Title: Between six towns Post by BMAD on May 23^{rd}, 2014, 3:08pm he smallest distance between any two of six towns is m miles. The largest distance between any two of the towns is M miles. Show that M/m > sqrt(3). Assume the land is flat. 

Title: Re: Between six towns Post by BMAD on May 31^{st}, 2014, 9:09pm Bump 

Title: Re: Between six towns Post by dudiobugtron on Jun 1^{st}, 2014, 4:07am One way to work this out would be to find the configuration which has the greatest ratio m:M, and show that even then it's not as great as 1:sqrt(3). I couldn't initially figure out what that best configuration would be, though, so I gave up! But since you bumped it, I'll think some more about it. 

Title: Re: Between six towns Post by pex on Jun 1^{st}, 2014, 4:45am Apparently sqrt(3) is not even a tight bound. According to this page (http://www2.stetson.edu/~efriedma/maxmin/) (spoiler alert: shows optimal configuration), the smallest possible value for M/m is [hide]sqrt( (5 + sqrt(5)) / 2 )[/hide] or approximately 1.90; sqrt(3) is approximately 1.73. Moreover, that page claims that the result is "trivial", but I have to admit I'm not seeing that. 

Title: Re: Between six towns Post by BMAD on Jun 1^{st}, 2014, 5:57am Wouldn't the sqrt (3) be tighter since it is smaller? 

Title: Re: Between six towns Post by pex on Jun 1^{st}, 2014, 6:31am on 06/01/14 at 05:57:19, BMAD wrote:
(By your reasoning, the bound M/m > 1 would be even tighter  but that's much easier to prove!) 

Title: Re: Between six towns Post by BMAD on Jun 1^{st}, 2014, 6:33am Oops. I got the inequality backwards in my head. 

Title: Re: Between six towns Post by Grimbal on Jun 1^{st}, 2014, 2:38pm I think the optimal configuration is a pentagon with one city in the center. The max distance is M=2*sin(2*pi/5) assuming m=1. 

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