the minimal diameter of a quadrilateral - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:55:13Z http://mathoverflow.net/feeds/question/85253 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85253/the-minimal-diameter-of-a-quadrilateral the minimal diameter of a quadrilateral GUBO 2012-01-09T13:08:29Z 2012-01-10T07:37:12Z <p>Let a convex quadrilateral ABCD with perimeter 1，d is the maximum of AB,AC,AD,BC,BD,CD,prove that d is not less than 1/3 we can prove that parallelogram ABCD with perimeter 1，than one of AC,BD is more than 1/3 but the general case is very difficult to solve.</p> http://mathoverflow.net/questions/85253/the-minimal-diameter-of-a-quadrilateral/85268#85268 Answer by ε-δ for the minimal diameter of a quadrilateral ε-δ 2012-01-09T16:53:26Z 2012-01-09T19:46:12Z <p><img src="http://www.freeimagehosting.net/newuploads/rgwvd.png" alt="Quadrilateral Image"></p> <p>It is not true.</p> <p>Take $A=0$, $B=1$, $C=e^{\frac\pi6 i}$ $D=e^{\frac\pi3 i}$. Then $d=1$ and the perimeter is $2+\tfrac1{\cos\frac{\pi}{12}} >3$.</p> <p>I am sure that $\frac1{2+\tfrac1{\cos\frac{\pi}{12}}}$ is the optimal bound.</p> http://mathoverflow.net/questions/85253/the-minimal-diameter-of-a-quadrilateral/85316#85316 Answer by David Eppstein for the minimal diameter of a quadrilateral David Eppstein 2012-01-10T07:37:12Z 2012-01-10T07:37:12Z <p>The answer given by ε-δ (a kite inscribed in a Reuleaux triangle) can be found in</p> <p>Ball, D. G. (1973), "A generalisation of π", <em>Mathematical Gazette</em> 57 (402): 298–303, doi:<a href="http://dx.doi.org/10.2307%252F3616052" rel="nofollow">10.2307/3616052</a>, JSTOR <a href="http://www.jstor.org/stable/3616052" rel="nofollow">3616052</a>;</p> <p>He doesn't give an explicit proof that this is optimal but says it can be done by "some tedious but not very difficult trigonometry. See also</p> <p>Griffiths, David; Culpin, David (1975), "Pi-optimal polygons", <em>Mathematical Gazette</em> 59 (409): 165–175, doi:<a href="http://dx.doi.org/10.2307%252F3617699" rel="nofollow">10.2307/3617699</a>, JSTOR <a href="http://www.jstor.org/stable/3617699" rel="nofollow">3617699</a></p> <p>for extensions to higher order polygons.</p>