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

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 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 ).