Stability theorems for projections of convex sets

LetH ^n?1 denote the set of all (n ? 1)-dimensional linear subspaces of euclideann-dimensional spaceE ⁿ (n≧3), and letJ andK be two compact convex subsets ofE ⁿ. It is well-known thatJ andK are translation equivalent (or homothetic) if for allH ∈H ^n?1 the respective orthogonal projections, sayJ _H, K_H, are translation equivalent (or homothetic). This fact gives rise to the following stability problem: Ifε≧0 is given, and if for everyH ∈H ^n?1 a translate (or homothetic copy) ofK _H is within Hausdorff distanceε ofJ _H, how close isJ to a nearest translate (or homothetic copy) ofK? In the case of translations it is shown that under the above assumptions there is always a translate ofK within Hausdorff distance (1 + 2√2)ε ofJ. Similar results for homothetic transformations are proved and applications regarding the stability of characterizations of centrally symmetric sets and spheres are given. Furthermore, it is shown that these results hold even ifH ^n?1 is replaced by a rather small (but explicitly specified) subset ofH ^n?1.