On Closed Sets With Convex Projections

On closed sets with convex projections under narrow sets of directions

“…Note that in these papers we are dealing with shadows in all directions. Remarkably, in this paper we show that the results in [5] 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. Thus we see that it suffices to have a 'narrow beam' of directions that produce convex shadows to find (n − 2)-manifolds in the sets C.…”

“…Since N ∩ C = ∅ and N ∩ B is bounded we can now show by precisely the same method as in the proof of [1,Theorem 3] that ψ N (0) / ∈ ψ N (C). Since 0 ∈ B we have that C is not a weak P-imitation of B, and the proof is complete.…”

“…Let ψ P * ( C ) = (P * ) ⊥ for some P * ∈ P and let ψ P (C) be convex The method used in [5] and [1] consists of finding a high-dimensional derived face of C of which it can be proved that its boundary is in C. This method can also be applied in the situation that P is a proper subset of L n n−k (see Theorem 16) but only if the face is consistent with P, by which we mean that it is contained in a supporting hyperplane H such that H + P = H for some P ∈ P. If such faces do not exist, then the method in [5] and [1] breaks down. We solve this problem by proving that if there are no high-dimensional faces of C that are consistent with P, then there exists a high-dimensional projection of C such that some open subset of its boundary can be 'lifted' back up to C; see the proof of Theorem 17.…”

“…Dijkstra, Goodsell, and Wright [5] improved on this result by showing that such a C must contain an (n−2)-sphere, so in this case projections cannot raise dimension by more than one. Barov, Cobb, and Dijkstra [1] were subsequently able to construct an extension of the result over the class of unbounded closed sets. Note that in these papers we are dealing with shadows in all directions.…”

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.…”