ON CLOSED SETS IN HILBERT SPACE WITH CONVEX PROJECTIONS UNDER SOMEWHERE DENSE SETS OF DIRECTIONS

Abstract

Let k be a fixed natural number. In an earlier paper the authors show that if C is a closed and nonconvex set in the Hilbert space ℓ2 such that the closures of the projections onto allk-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of ℓ2. Here this theorem is strengthened significantly by making the much weaker assumption that the set of projection directions is somewhere dense. To show the sharpness of the main theorem we construct "minimal imitations" of closed convex sets in ℓ2. In addition, we show that closed convex sets with an empty geometric interior cannot be imitated by other closed sets.