Stability theorems for projections of convex sets (Q1098104)

Several results and problems of convexity have been considered in recent years under the aspect of stability. The author gives the following interesting theorem: Let C,D be two non-empty compact convex subsets of \({\mathbb{E}}^ d\). Let \(\epsilon >0\) and assume that for any hyperplane H the orthogonal projection of C into H has (Hausdorff) distance \(\leq \epsilon\) from a suitable translate of the orthogonal projection of D into H. Then C has distance \(\leq (1+2\sqrt{2})\epsilon\) from a suitable translate of D. The constant \(1+2\sqrt{2}\) is best possible, but may be replaced by 1 if C,D are both centrally symmetric. Several refinements, applications and related results concerning homotheties instead of translations are given.