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Hej, there seem to be a number of variations of the original question out there. Let me tell you what is known (to me):

Let us assume that an (unknown) convex polytope $P$ in $R^3$ is given. I assume that we can observe orthogonal projections onto planes (think of shadows that we see on a screen that is orthogonal to the direction of the incoming parallel light). If the direction of the incoming light is a unit vector u, then the information we have is the data $F(P,u)=area(proj_u(P))$ where $proj_u(P)$ is the orthogonal projection of $P$ along the direction $u$. This function is called projection function or brightness function in convex geometry. What do we know about $P$, when we know the projection function for all directions u?

1) The surface area of $P$ is proportional to the averaged function $F(P,u)$, averaged over all directions (was remarked before, and is a consequence of Cauchy's projection formula, even true in all dimensions and when the polytope is some convex set).

2) If $P$ is translated then $F(u)$ does not change, so the best we can hope for is determination up to translations. In general $P$ is not uniquely determined up to translations by $F(P,u)$. Also this was remarked earlier. It is also clear. Take some polytope $P$ which does not have a centre of symmetry. Then its reflection at the origin $\hat P$ is not a translation of the original $P$, but $F(P,u)=F(\hat P,u)$ for all $u$.

3) Amazingly, the central symmetry is the crucial property: If $P$ is central symmetric the $F(P,u)$ (known for all u) determines $P$ uniqely up to translations. This was already shown by Alexandrov and is often called Alexandrov's projection theorem. It holds in all dimensions and for arbitrary central symmetric convex bodies. See Richard Gardner's book on geometric tomography.

4) Under the central symmetry assumption, there is an algorithm to determine $P$ from $F(P,u)$, see the article RECONSTRUCTION OF CONVEX BODIES FROM BRIGHTNESS FUNCTIONS by R. Gardner and Peyman Milanfar.

Best wishes, Markus Kiderlen