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

Abstract

Abstract. Let k, n ∈ N with k < n and let G k (R n ) denote the Grassmann manifold consisting of all k-dimensional linear subspaces in R n . In an earlier paper the authors showed that if the projections of a nonconvex closed set C ⊂ R n are convex and proper for projection directions from some nonempty open set P ⊂ G k (R n ), then C contains a closed copy of an (n−k −1)-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that P is somewhere dense in G k (R n ).

On closed sets with convex projections under narrow sets of directions

“…In a forthcoming paper [2] we show that in Theorems 1, 18, and 19 the premise that P is open can be relaxed to the condition that P ⊂ int P.…”

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

“…We end this section with the proof of Theorem 4 for which we need the following results from [4,Lemma 11] Theorem 22. Let k ∈ N with k < dim V, let B be a convex and closed set in V, and let P be a subset of G k (V) such that P ⊂ int P. If C is a closed set that is a weak P-imitation of B, then E k (B, P) ⊂ C.…”

“…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.…”

“…In order to prove Theorem 1, we introduce the set E k (B, P) consisting of P-extremal points of B; see Definition 4. We proved in [4,Theorem 15] that every closed weak P-imitation of B contains the set E k (B, P); see Theorem 22. Theorem 1 is then proved by finding the copy of 2 in the set E k (B, P); see Theorem 25.…”

A Cantor set in Rd with “large” projections

“…For example, given a topological space X and integers d > m 0, is there an embedding j : X → R d such that for each m-plane, the projection of j( X) onto it is convex? For this setting, see [8, Theorem 3; Example, p. 124], [10,3], and joint papers of S. Barov and J.J. Dijkstra (see [4] and references therein; also for the case of Hilbert space). 5) Many papers of the last decades are devoted to embeddings which have other very special properties relative to all m-dimensional affine planes and in particular, to all affine straight lines.…”