Normal bundles of convex bodies

The family of normal bundle cones is a meager subset of the space of all closed convex cones in Ed2. To find out whether a cone is a normal bundle cone, a simple criterion is given. The dimension of a normal bundle cone NC is between 12d(d+1) and d2. The lower bound is attained precisely for ellipsoids. As the dimension of NC increases, the ellipsoidal character of the convex body C and the family of linear vector fields which are tangent on the boundary of C decrease. For generic C the dimension of NC is d2. For d=2,3 a complete description of the situation is given. Next, symmetry properties are studied. The cone NC coincides with its polar, its polar transpose, or its transpose, if and only if C is a disc with rotational symmetry of order 4, a Radon disc, a Euclidean ball, or an ellipsoid. A conjecture deals with isometries and linear automorphisms of NC and C. A relation between symmetry properties of a pair of normal bundle cones and orthogonality in two normed spaces is stated. There is a bijection between the family of certain faces of NC and the family of planar shadow boundaries of C with respect to parallel illumination. A conjecture deals with a more general case.