On closed sets with convex projections under narrow sets of directions

Abstract

Abstract. Dijkstra, Goodsell, and Wright have shown that if a nonconvex compactum in R n has the property that its projection onto all k-dimensional planes is convex, then the compactum contains a topological copy of the (k − 1)-sphere. This theorem was extended over the class of unbounded closed sets by Barov, Cobb, and Dijkstra. We show that the results in these two papers remain valid under the much weaker assumption that the collection of projection directions has a nonempty interior.

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

“…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.…”

“…Observe that [3,Theorem 18] corresponds to Theorem 1 with the additional assumption that P is open in G n−k (R n ). 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.…”

“…Observe that [3,Theorem 18] corresponds to Theorem 1 with the additional assumption that P is open in G n−k (R n ). 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 .…”

“…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.…”

“…Also, a k-plane in R n is an affine subspace of R n of dimension k. Now, let B be a closed convex subset of R n that contains no k-plane and let P ⊂ G n−k (R n ) be open such that B is not a P-imitation of R n . In this setting the authors showed in [3,Theorem 18] that if C is a closed weak P-imitation of B with C = B, then C ∩ B contains a closed copy of a (k − 1)-manifold. The main purpose of this paper is to show that the above result remains valid if we weaken the assumption on P from open to somewhere dense in G n−k (R n ).…”