Convex Bodies with Affinely Equivalent Projections and Affine Bodies of Revolution

We study affine bodies of revolution. This will allow us to prove that a convex body all whose orthogonal n-projections are affinely equivalent is an ellipsoid, provided n ≡ 0, 1, 2 mod 4, n > 1, with the possible exemption of n = 133. Our proof uses convex geometry and topology of compact Lie groups.