What is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape? 
 A: May be you can generalize it by considering a convex figure to be made of infinitely many infinitely small squares.
If you generalize that the average area of the shadow of that square is 1/2 the area of one side of square you can say that for any convex figure the average area of shadow would be 1/4 the total area of figure.
If you are wondering that a square sheet we are considering is giving the average area 1/2 while the figure is giving 1/4.Thats because in any orientation half of the figure will not be responsible for casting the shadow (like in shpere if you cut the sphere in half parallel to the plane it is casting it's shadow on, the sphere will still cast a shadow or area πr^2 in that orientation) that's why at any instant only half of the total shadow of individual square plates will be responsible for shadow of the figure.
Now you want to prove that the average area of the square plate is 1/2 its area then you can prove it with help of a sphere by considering it to be made up of infinitely many infinitely small squares and as the are all symmetric and are equally contributing to the area then you can say that each of them will  cast an average area of 1/2 their area of one side.
Sorry for the bad English and explanations, I am not too good at it.
