Some geometric properties of typical compact convex sets in Hilbert spaces

An investigation is carried out of the compact convex sets X in an infinite-dimensional separable Hilbert space E, for which the metric antiprojection q(X)(e) from e to X has fixed cardinality n + 1 (n is an element of M arbitrary) for every e in a dense subset of IE. A similar study is performed in the case of the metric projection p(X)(e) from e to X where X is a compact subset of E.