The structure of typical compact sets in Euclidean space (Q1972627)

Let \(E^n\) be the \(n\)-dimensional Euclidean space \((n\geq 2)\). The author endows the set of nonempty compact sets in \(E^n\) with the Hausdorff metric and considers a typical (in the sense of Baire category) compact set \(C\in E^n\). He proves that (1) \(C\) is homeomorphic to a perfect Cantor set; (2) the convex hull of \(C\) is a smooth convex body; (3) the set of extreme points of the convex hull of \(C\) is homeomorphic to a perfect Cantor set; (4) the set of distances realized on \(C\) is homeomorphic to a perfect Cantor set.