On closed sets with convex projections in Hilbert space

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

“…Dijkstra, Goodsell, and Wright [8] improved on this result by showing that such a C must contain an (n − 2)-sphere. Barov, Cobb, and Dijkstra [1] were subsequently able to construct an extension of that result over the class of unbounded closed sets and [2] concerns the Hilbert space variant of the problem. In [3] we showed that the results in [8] and [1] remain valid if we make the much weaker assumption that the collection of projection directions that produce convex shadows has a nonempty interior.…”

“…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.…”

“…Choose a coordinate system for V such that x = 0. By assumption there are distinct supporting hyperplanes H 1 , H 2 to B at 0, and there are v 1 , v 2 Select a basis {e 1 , . [{e 1 , . . .…”

On Closed Sets in Hilbert Space With Convex Projections Under Somewhere Dense Sets of Directions

“…Barov et al [1] have shown that if all projections onto k-planes, 1 ≤ k ≤ n − 1, of C are convex and proper in a significant number of directions, then C contains a closed subset that is (k−1)-manifold without boundary. Subsequently, the authors have shown in [2] that if C is a closed and nonconvex set in the Hilbert space 2 such that the closures of S. Barov & J. J. Dijkstra the projections onto all k-hyperplanes (planes with codimension k) are convex and proper then C must contain a closed copy of 2 . Moreover, in [3,4] the authors show that the above result in [1] remains valid if we make much weaker assumption that the collection of projection directions that produce convex projections is somewhere dense.…”

“…Having all this in mind, we naturally ask ourselves whether the results in Hilbert space, obtained in [2], are valid if we require convexity of the projections only for a somewhere dense set of directions. So the main purpose of this paper is to give a positive answer to that question and to generalize the Imitation Theorem [2,Theorem 2], that is, to find "minimal imitations" of closed and convex sets for arbitrary sets of projection directions.…”