The convex hull of a typical compact set
The space C of all nonempty compact subsets of \({\mathbb{R}}^ d(d\geq 2)\) with the Hausdorff metric is a Baire space, and a property is said to be typical in C or to hold for most \(K\in C\) if it holds for all \(K\in C\) except those in a set of first Baire category. In this note, typical properties of compact sets concerning the boundary structure of their convex hull are studied. It turns out, e.g., that the set of extreme points is homeomorphic to Cantor's ternary set E, the set of exposed points is homeomorphic to the space of irrational numbers and the set of farthest points is homeomorphic to the topological product \({\mathbb{Q}}\times E\), where \({\mathbb{Q}}\) is the space of rational numbers. The paper contains also some results concerning the relative position of these boundary sets, the smoothness of the boundary and the dimension of the p-skeleton. While the convex hull of a typical compact set is not a typical convex body, the last theorem shows that for a typical convex body the set of farthest points is also homeomorphic to \({\mathbb{Q}}\times E\).
scientific article; zbMATH DE number 4036603
scientific article