TITLE: Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections
QUESTION [9 upvotes]: My general question concerns what we can learn about an arbitrary, three-dimensional convex polytope (or convex hull of an arbitrary polytope) strictly from the surface areas of its two-dimensional projections on a plane as it 'tumbles' in 3-space (i.e. as it rotates along an arbitrary, shifting axis).  
If it's helpful, please imagine the following physical set-up:
We take an arbitrary three-dimensional convex polytope, and fix the center of mass to a coordinate in 3-space, $(x_0, y_0, z_0)$, located at some distance, $D$, above a flat surface.  While this prohibits translation of the center of mass, the polytope is still allowed to tumble freely (i.e. it is allowed rotation around an arbitrary axis centered at the fixed coordinate).
There is no 'gravity' or other force to stabilize the tumbling polytope in a particular orientation.  Over time it will continue to tumble randomly.  (The 'physical' set-up is only meant for descriptive reasons.)
We shine a beam of coherent light on the tumbling polytope, larger than the polytope's dimensions, and continually record the area of the resulting shadow, or two-dimensional projection on the surface.  To be clear, the area of the two-dimensional projection is the only information we are allowed to observe or record, and we are allowed to do so over an arbitrary length of time.

My question is - From observing the area of the tumbling polytope's shadow, or two-dimensional projection over time, what can we learn about it's geometry?  To what extent can we characterize and/or reconstruct the polytope from its changing shadow (extracting the surface area for example - hat tip to Nurdin Takenov)? 
Do we gain anything by watching the evolution of the convex polytopes shadow as it tumbles (part of the point for the physical example), as opposed to an unordered collection of two-dimensional projections?
Update - Nurdin Takenov (and Sergei Ivanov in later comments) nicely points out that we can use the average surface area of the two-dimensional projection to find the surface area of the tumbling convex polygon.  Might we be able to find it's volume?  
(Addendum - I would be really neat if somebody could point me to any algorithms in the literature... or available software.... that let's me calculate and characterize two-dimensional surface projections of convex polytopes!)

REPLY [6 votes]: 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