Answer by Anton Petrunin for Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?

The first two questions (and others) are discussed in the following two papers of Kuzminykh, A. V.:

• An isoprojection property of the sphere (1973)
• Reconstructibility of a convex body from the set of its projections (1984)