A conjecture of parallelogram inside convex and central symmetric curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:24:36Z http://mathoverflow.net/feeds/question/44858 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44858/a-conjecture-of-parallelogram-inside-convex-and-central-symmetric-curve A conjecture of parallelogram inside convex and central symmetric curve galois 2010-11-04T18:24:50Z 2012-03-09T16:27:47Z <p>Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$.</p> <p>How to prove the conjecture that $\displaystyle \frac{S'}{S} \ge \frac{2}{\pi}=0.6366\dots$?</p> <p>For example, If Q is an ellipse, $\displaystyle S'=2ab$, $\displaystyle S=\pi ab$. If Q is a regular hexagon, $\displaystyle \frac{S'}{S}= \frac{2}{3}$.</p> <p>It's trivial that $\displaystyle \frac{S'}{S} \ge \frac{1}{2}$, and I know how to prove $\displaystyle \frac{S'}{S} \ge \frac{4}{4+\pi}=0.56\dots$</p> <p>From many reason, I believe this conjecture is true. Denote MAP="Maximum Area Parallelogram": For any Q and any direction $\theta$, let $P(Q,\theta)$ be the area of MAP which have a corner in this direction. $S'=max{P(Q,\theta)}$. In order to make $\frac{S'}{S}$ smallest, We need keep the largest one of ${P(Q,\theta)}$ small while S is a constant. Ellipse just keeps everyone in ${P(Q,\theta)}$ average. This is very special, I don't think there will be other curve having this property. On the other hand, distribute equally always lead to the min-max in our knowledge.</p> <p>About the $\frac{4}{4+\pi}$ lowerbound, the idea is as follows: First, use polar function $r(\theta)$ to describe the curve. The condition is that $r(a)*r(b)*sin|a-b|&lt;=C$, and we want to bound is $S=\Integral_{\theta}{r(\theta)}^2$. Second, Without lose of generality, We assume $r(0)=r(90)=1,C=1$, and assume $Q$ is in the boundary of $Z={ (x,y)|-1\le x,y\le 1}$. Third, let $a=r(\theta)$ and $b=r(\theta+90)$ and find a bound for $(a^2+b^2)$ by Cauchy-Inequality. and it will give a bound for the area $S$.</p> http://mathoverflow.net/questions/44858/a-conjecture-of-parallelogram-inside-convex-and-central-symmetric-curve/90725#90725 Answer by alvarezpaiva for A conjecture of parallelogram inside convex and central symmetric curve alvarezpaiva 2012-03-09T16:18:19Z 2012-03-09T16:27:47Z <p>This is an old result in convex geometry. See</p> <p>E. Sas, ¨Uber ein Extremumeigenschaft der Ellipsen, Compositio Math. 6 (1939) 468– 470.</p> <p>A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951) 245–247.</p> <p>Fedor Petrov's remark was right on target. Steiner symmetrization gives an easy proof (see Macbeath). </p> <p>By the way, the equality case holds only for ellipses, which explains the titles of the papers above.</p>