Typical convex sets (Q1594588)

Let \(K\) be a compact convex subset of a Hilbert space \(\mathcal B\) and let \(\mathcal B (K)\) be the set of compact convex subsets of \(K\) equipped with the Hausdorff metric. A typical subset of a metric space is the complement to the union of a countable family of nowhere dense sets. In [J. Aust. Math. Soc., Ser. A 43, 287-290 (1987; Zbl 0629.52002)], \textit{T. Schwarz} and \textit{T. Zamfirescu} proved that the typical sets in the cone of compact convex subsets of the Euclidean space are the sets in which the extreme points are typical. The author extends this result to the general situation and establishes some properties of typical elements of the set \(\mathcal B (K)\). He proves that if \(K\) is an infinite-dimensional set then nowhere dense compact convex sets in \(K\) of zero co-dimension and with typical extreme points are typical in \(\mathcal B (K)\). The situation changes if \(K\) is finite-dimensional. The author proves that, in this case, the sets \(U\in\mathcal B (K)\) of full dimension and with smooth boundary coinciding with \(\text{ext} U\) are typical in \(\mathcal B(K)\).