While driving back home I heard an interesting math question from all places, the Car Talk show on NPR. This made me think of a generalization of the trick they used and in particular, formulate the following problem. $\newcommand{\bR}{\mathbb{R}}$
Problem Determine all, reasonably well behaved compact domains $D\subset \bR^2$ with the following property: any line through the origin divides $D$ into two regions of equal areas. We will refer to this as property $\boldsymbol{C}$ (for cut).
I know that "reasonably well behaved" is a rather fuzzy requirement. At this moment I don't want to think of Cantor like weirdos. So let's assume that $D$ is semialgebraic.
We say that a domain $D$ satisfies property $\boldsymbol{S}$ (for symmetry) if it is invariant with respect to the involution
$$\bR^2\ni (x,y)\mapsto (-x,-y) \in \bR^2. $$
It is not hard to see that
$$\boldsymbol{S}\Rightarrow \boldsymbol{C}. $$
Is the converse true?
I describe below one situation when this happens.
A special case. I'll assume that $D$ is semialgebraic, star-shaped with respect to the origin and satisfies $\boldsymbol{C}$. We can then describe $D$ in polar coordinates by an inequality of the form
$$(r,\theta)\in D \Longleftrightarrow 0\leq f(\theta),\;\;\theta\in[0,2\pi], $$
where $f:[0,2\pi]\to (0,\infty)$ is a semialgebraic function such that $R(0)=R(2\pi)$. We can extend $f$ by $2\pi$-periodicity to a function $f:\bR\to [0,\infty)$ whose restriction to any finite interval is semialgebraic.
For any $\phi\in[0,2\pi]$ denote by $\ell_\phi:\bR^2\to \bR$ the linear functiondefined by
$$\ell_\phi (x,y)= x\cos\phi +y\sin\phi. $$
Denote by $A(\phi)$ the area of the region $D\cap \bigl\{ \ell\phi\geq 0\bigr\}$. Since $D$ satisfies $\boldsymbol{C}$ we deduce
$$A(\phi)=\frac{1}{2} {\rm area}\;(D), $$
so that
$$A'(\phi)=0,\;\;\forall \phi. $$
Observe that
$$ A(\phi+\Delta \phi)-A(\phi) =\int_{\phi+\frac{\pi}{2}}^{\phi+\frac{\pi}{2}+\Delta\phi} \left(\int_0^{f(t)} r dr\right) dt -\int_{\phi+\frac{3\pi}{2}}^{\phi+\frac{3\pi}{2}+\Delta\phi} \left(\int_0^{f(t)} r dr\right) dt. $$
For simplicity we set $\theta=\theta(\phi)=\theta+\frac{\pi}{2}$. We can then rewrite the above equality as
$$ A(\phi+\Delta \phi)-A(\phi)=\int_\theta^{\theta+\Delta\theta}\left(\int_0^{f(t)} r dr\right) dt -\int_{\theta+\pi}^{\theta+\pi+\Delta\theta} \left(\int_0^{f(t)} r dr\right) dt. $$
Hence
$$0=A'(\phi)= \frac{}{2}\Bigl( f(\theta)^2-f(\theta+\pi)^2\Bigr). $$
Hence $f(\theta)= f(\theta+\pi)$, $\forall \theta$. This shows that $D$ satisfies the symmetry condition $\boldsymbol{S}$.
$$ \ast\ast\ast $$
Here is a simple instance when $\boldsymbol{C}$ does not imply $\boldsymbol{S}$.
Suppose that $D$ is semialgebraic and has the annular description
$$ f(\theta)\leq r\leq F(\theta). \tag{1}\label{1}$$.
Using the same notations as above we deduce that
$$ 0=A'(\phi)= \frac{1}{2} \Bigl(F^2(\theta)-f^2(\theta)\Bigr)- \frac{1}{2} \Bigl(F^2(\theta+\pi)-f^2(\theta+\pi)\Bigr). $$
Thus, the domain (\ref{1}) satisfies $\boldsymbol{C}$ iff the function $ G(\theta)=F^2(\theta)-f^2(\theta)$ is $\pi$-periodic. Note that
$$F(\theta)= \sqrt{f^2(\theta)+ G(\theta)}.$$
If we choose
$$f(\theta)= e^{\sin \theta},\;\; G(\theta)=e^{\cos 2\theta},$$
then we obtain the domain bounded by the two closed curves in the figure below. This domain obviously violates the symmetry condition $\boldsymbol{S}$.
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