4 Mention a new paper by Gardner et al.

This is a nice question, and I look forward to seeing a definitive answer. Meanwhile, let me point out that it is known that a convex polytope in $E^n$ is not always determined by a finite set of its projections, but it is determined by all of its 2D projections. Richard Gardner, in Geometric Tomography, puts the first point this way: "it is generally not possible to choose a finite set of subspaces in such a way that the corresponding projections distinguish $P$ from every other convex polytope" [p.93].

For some reason I cannot add a comment to reply to Rob Grey's latest. My apologies for not understanding the MathFlow conventions. Here is my answer:

Yes, I wonder also. That's why what I posted is not an answer to your interesting question. You have rather less info--just the area, not the actual projection--but you have the "evolution" as you put it, presumably an association of the direction of projection with the area. This could be viewed as an area associated with each point on the unit sphere (the point representing the projection direction). I guess here I am presuming a particular answer to Sergei Ivanov's question.

So I would rephrase your question: Does the map from $S^2$ to projection area uniquely determine the polytope, or more generally, a convex body?

There is a new paper on the arXiv by Gardner, Gronchi, and Theobald that addresses a very specific case of Rob's original question: "Determining a rotation of a tetrahedron from a projection" arXiv:1111.7100. They determine the conditions under which the (orthogonal projection) shadow of a known tetrahedron permits reconstructing its orientation in space. This is already a richly complex situation.

3 typo

This is a nice question, and I look forward to seeing a definitive answer. Meanwhile, let me point out that it is known that a convex polytope in $E^n$ is not always determined by a finite set of its projections, but it is determined by all of its 2D projections. Richard Gardner, in Geometric Tomography, puts the first point this way: "it is generally not possible to choose a finite set of subspaces in such a way that the corresponding projections distinguish $P$ from every other convex polytope" [p.93].

For some reason I cannot add a comment to reply to Rob Grey's latest. My apologies for not understanding the MathFlow conventions. Here is my answer:

Yes, I wonder also. That's why what I posted is not an answer to your interesting question. You have rather less info--just the area, not the actual projection--but you have the "evolution" as you put it, presumably an association of the direction of projection with the area. This could be viewed as an area associated with each point on the unit sphere (the point representing the projection direction). I guess here I am presuming a particular answer to Sergei Ivanov's question.

So I would rephrase your question: Does the map from $S^2$ to projection area uniquely determine the polytope, or more generally, a convex body?

2 Looked up Gardner.

This is a nice question, and I look forward to seeing a definitive answer. Meanwhile, let me point out that it is known that a convex polytope in $E^n$ is not always determined by a finite set of its projections, but it is determined by all of its 2D projections. I think this is Richard Gardner, in Gardner's Geometric Tomography., puts the first point this way: "it is generally not possible to choose a finite set of subspaces in such a way that the corresponding projections distinguish $P$ from every other convex polytope" [p.93].

1