Suppose that S is a compact convex subset of the Euclidean plane E whose interior is nonempty. 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 centrosymmetric with respect to p?

6$\begingroup$ Yes. Take a look at a previous MO thread mathoverflow.net/questions/32690/… $\endgroup$– Andrey RekaloJun 15 '11 at 22:50
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 centersymmetric. $$*$$
Details: Assume $p=0\in\mathbb{C}$ and let $H _ +$ and $H _ $ be resp. the upper and lower halfplanes. 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 $p$ ("starshaped"). If we also drop the condition of starshapeness, it is easy to make nonsymmetric counterexamples: e.g. the set of all $z$ with $z\le 4$ in the upper halfplane and all $z$ with $3\le z\le 5$ in the lower halfplane.

1$\begingroup$ I see now the thorough answer by Andrey Rekalo to a similar previous question; I'm not deleting mine as it contains some hint) $\endgroup$ Jun 15 '11 at 23:14

$\begingroup$ Thanks for your answers and all the helpful information they contain. $\endgroup$ Jun 15 '11 at 23:32