Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose that S is a compact convex subset of the Euclidean plane E whose interior is non-empty. If p is a point of E such that every straight line in E which passes through p bisects the area of S, is S necessarily centro-symmetric with respect to p?

share|improve this question
5  
Yes. Take a look at a previous MO thread mathoverflow.net/questions/32690/… –  Andrey Rekalo Jun 15 '11 at 22:50
add comment

1 Answer

up vote 2 down vote accepted

Yes: the condition is equivalent to : any straight line through $p$ intersects $S$ in a segment whose midpoint is $p$, that is the convex is center-symmetric. $$*$$

Details: Assume $p=0\in\mathbb{C}$ and let $H _ +$ and $H _ -$ be resp. the upper and lower half-planes. Consider the difference of the areas of the two halves of $S$ cut by a rotating line: $$\delta(\alpha):=\operatorname {Area}(S\cap e^{i\alpha}H _ +) - \operatorname {Area}(S\cap e^{i\alpha}H _ -) $$ The derivative of this quantity wrto $\alpha$ is $$\delta'(\alpha)=\frac{1}{2}\operatorname {Length}^2(S\cap e^{i\alpha}\mathbb{R} _ -) - \frac{1}{2}\operatorname {Length}^2(S\cap e^{i\alpha}\mathbb{R} _ +), $$ as it immediately follows writing the areas as integrals in polar coordinates. In particular, $\delta(\alpha)$ is constant iff $p$ is the midpoint of $S\cap e^{i\alpha}\mathbb{R}$.

Remark: the same argument holds of course if we only assume $S$ to be convex with respect to $p$ ("star-shaped"). If we also drop the condition of star-shapeness, it is easy to make non-symmetric counterexamples: e.g. the set of all $z$ with $|z|\le 4$ in the upper half-plane and all $z$ with $3\le |z|\le 5$ in the lower half-plane.

share|improve this answer
1  
I see now the thorough answer by Andrey Rekalo to a similar previous question; I'm not deleting mine as it contains some hint) –  Pietro Majer Jun 15 '11 at 23:14
    
Thanks for your answers and all the helpful information they contain. –  Garabed Gulbenkian Jun 15 '11 at 23:32
add comment

Your Answer

 
discard

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.