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.

On closed sets with convex projections under somewhere dense sets of directions

“…The method we use is to show that if P satisfies the premises of Theorem 1, then int P contains a nonempty open subset that satisfies the premises of [3,Theorem 18]; see Theorem 13. Let us point out that our approach for proving the reduction of Theorem 1 to [3,Theorem 18] is sufficiently general so as to include the case that the ambient space is the separable Hilbert space 2 so that the results are also of use for a forthcoming extension [4] of the results in [2] and [3] over 2 . Theorem 1 deals with the retrieval of information about a geometric object from data about its projections, which places the result in the field of Geometric Tomography; see Gardner [10] for background information.…”