Properties of typical bounded closed convex sets in Hilbert space

F S De Blasi,N V Zhivkov

doi:10.1155/aaa.2005.423

Abstract

For a nonempty separable convex subsetXof a Hilbert spaceℍ(Ω), it is typical (in the sense of Baire category) that a bounded closed convex setC⊂ℍ(Ω)defines anm-valued metric antiprojection (farthest point mapping) at the points of a dense subset ofX, whenevermis a positive integer such thatm≤dimX+1.

Highlights

Baire category techniques are known to be a powerful tool in the investigation of the convex sets
We consider some geometric properties of typical 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

Summary

Introduction

Baire category techniques are known to be a powerful tool in the investigation of the convex sets. 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. If dim X ≥ m, there exists a residual set ᏾c ⊂ Ꮿ such that for C ∈ ᏾c the set LmX (C) contains an everywhere continual in X subset at each point of which Q(·,C) is upper semicontinuous

Notation

Topological facts

Lemmas

Main construction