Uniqueness and stability of the solution of a problem of geometry in the large

This paper considers the problem of determining a convex surface from the area $F(n)$ of its orthogonal projection on any plane $(x,n)=0$ and the area $S(n)$ of the portion of the surface illuminated in the direction $n$. It is proved that in a certain class a convex surface is uniquely defined (up to translation) by a function $\varphi(n)=2aF(n)+bS(n)$ for $a\ne0$, $b\ne0$, $a+b\ne0$. Moreover, the surface is analytic if and only if $\varphi(n)$ is an analytic function on the unit sphere. The surface is shown to be stable, and a quantitative estimate related to stability is given.
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