This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and What can be said about 2 convex solids with corresponding maximal planar sections having equal area? and could be of interest if the answers to the questions raised in both those posts are negative.
Let there be two convex solids P and Q that hover over the XY plane satisying: areas of shadows of both P and Q on the XY plane are equal and areas of maximal area cross sections of both P and Q that are parallel to XY plane are also equal. Now, if the same rotations are done on both P and Q keeping their centers of mass fixed and both areas areas of shadow and max cross section) of P remain equal to the corresponding areas of Q for any same rotation done to both P and Q, what can one conclude about P and Q - will they be congruent/have equal volume/equal surface area?
Same question as above with 'perimeter' replacing 'area'.
If congruence cannot be guaranteed in either case above, will both equality of areas and equality of perimeters holding be sufficient?
