Abstract
Roughly speaking, in this article we prove quantitative versions of the following statements: (i) If each shadow boundary of a convex body in E d (d≥ 3) under parallel illumination contains a curve which is almost planar and has the same shadow as the shadow boundary, then the body is approximately ellipsoidal. (ii) If for each diameter of a convex disk in E 2 there is a diameter which is not far from being conjugate to the former, then the disk is close to a Radon disk. A consequence of these theorems is a stability result for the symmetry of orthogonality in finite-dimensional normed spaces. In addition, a stability result for the relation between different definitions of length in a normed plane is given.
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