[A recent episode of Star Talk Radio](https://www.startalkradio.net/show/exploring-exoplanets-seth-shostak-startalk-stars/ "Link to podcast episode") discussed among other things the unknown object(s) orbiting [Tabby's star](https://en.wikipedia.org/wiki/KIC_8462852 "Wikipedia entry") (aka "Alien mega structure discovered!" in non-scientific media) and an astronomer said with confidence that we know that whatever is orbiting that star isn't a planet because "it isn't round". Given that the mysterious object(s) were discovered by examining the variations in luminosity coming from the star, this got me thinking about light curves and what they can tell us about transiting objects.


Most generally my question is this:



> 
> Given a closed subset $A\subseteq D\subseteq\mathbb{R}^2$ where $D$ is the closed unit disc, what can we infer about the shape of $A$ when we are given the "light curve" $L\_A(t)=\lambda^2(D\setminus (A+te\_1))$ or equivalently if we are given $t\mapsto\lambda^2(D\cap (A+te\_1))$.
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In particular I would like to know:



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> Can we distinguish convex sets from non-convex sets? And if yes, are convex $A$ uniquely determined by their $L\_A$ (up to obvious non-uniquenesses like reflection across the x-axis)?
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Doing some elementary geometry one finds that circles are indeed detected by their light curves thus validating Dr. Shostak's claim.



Infering shapes from overlap with a shifting circle
A convex set and a non-convex set can have the same light curve.


Let $r$ be the reflection about the $x$-axis. Then $r(S)$ has the same light curve as $S$. So, let $C$ be a convex set above the x-axis, and let $C = C\_1 \cup C\_2$. Then $C$ has the same light curve as $C\_1 \cup r(C\_2)$. 


