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Let and be two convex bodies in with countably many diameters, such that their projections onto all 4 dimensional subspaces containing one fixed diameter are directly congruent. We show that if these projections have no rotational symmetries, and the projections of on certain 3 dimensional subspaces have no symmetries, then up to a translation. We also prove the corresponding result for sections of star bodies.
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M. Angeles Alfonseca, Michelle Cordier, Dmitry Ryabogin, On bodies in with directly congruent projections or sections. Rev. Mat. Iberoam. 35 (2019), no. 6, pp. 1745–1762DOI 10.4171/RMI/1100