On bodies with directly congruent projections and sections

Original · EN

Let K and L be two convex bodies in R⁴, such that their projections onto all 3-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfy an additional condition and some projections do not have certain symmetries, then K and L coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the n-dimensional versions of these results.