Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$.

How to prove the conjecture that $\displaystyle \frac{S'}{S} \ge \frac{2}{\pi}=0.6366\dots$?

For example, If Q is an ellipse, $\displaystyle S'=2ab$, $\displaystyle S=\pi ab$. If Q is a regular hexagon, $\displaystyle \frac{S'}{S}= \frac{2}{3}$.

It's trivial that $\displaystyle \frac{S'}{S} \ge \frac{1}{2}$, and I know how to prove $\displaystyle \frac{S'}{S} \ge \frac{4}{4+\pi}=0.56\dots$

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.

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 $r(a)r(b)\sin|a-b|\leq C$, and we want to bound is $S=\int_{\theta}{r(\theta)}^2$. Second, Without lose of generality, We assume $r(0)=r(90)=1,C=1$, and assume $Q$ is in the boundary of $Z=\{(x,y)|-1\leq x,y\leq 1\}$.
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 $S$.

share|cite|improve this question
What are some of the "many reasons" ? – futurebird Nov 4 '10 at 16:55
Also, if you could edit the question with a brief sketch of proof of the 4/(4+pi) bound, it might help others modify it to give better bounds. – Aryabhata Nov 4 '10 at 17:03

You might find this paper of interest:

Fulton C.M., Stein S.K. (1960) Parallelograms inscribed in convex curves. Amer Math Monthly 67: 257–258

share|cite|improve this answer
Thank you Joseph Malkevitch! This paper is helpful for my research. But for this specific problem, I find the paper doesn't help much. – galois Nov 5 '10 at 5:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.