Properties of Typical Bounded Closed Convex Sets in Hilbert Space

Baire category techniques are known to be a powerful tool in the investigation of the convex sets. Their use, which goes back to the fundamental contribution of Klee [17], has permitted to discover several interesting unexpected properties of convex sets (see Gruber [14], Schneider [23], Zamfirescu [25]). A survey of this area of research and additional bibliography can be found in [15, 27]. In the present paper, we consider some geometric properties of typical (in the sense of the Baire categories) nonempty bounded closed convex sets contained in a separable real Hilbert space. It will be shown in the typical case, for a closed convex and bounded set C and an integer m, that there is a dense subset D of the Hilbert space H such that the farthest point mapping generated by C is precisely m-valued at the points of D. A result of this type was recently obtained in [5], for typical nonempty compact convex sets. However, the approach of [5] cannot be adopted here for, in absence of compactness, the antiprojection mapping could have empty images. To overcome this difficulty we will use some ideas from [28], developed in the framework of the metric projections. Throughout, H(Ω) is a Hilbert space over the field of real numbers R whose elements are mappings x : Ω→Rwith countably many nonzero values and convergent sums ∑ ω∈Ω x2 ω. We often prefer to denote H(Ω) by H. It is assumed always in the paper that dimH(Ω) ≥ 2. As usual, the inner product and the norm are denoted by 〈·,·〉 and | · |. For a nonempty bounded set M ⊂H, the function f (x,M) = sup{|x− z| : z ∈M} is the farthest distance function, and the set-valued mapping