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

Abstract:

Let $k,n\in \mathbb {N}$ with $k<n$ and let ${\mathcal G}_k(\mathbb {R}^n)$ denote the Grassmann manifold consisting of all $k$-dimensional linear subspaces in $\mathbb {R}^n$. In an earlier paper the authors showed that if the projections of a nonconvex closed set $C\subset \mathbb {R}^n$ are convex and proper for projection directions from some nonempty open set $\mathcal {P}\subset {\mathcal G}_{k}(\mathbb {R}^n)$, 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 $\mathcal {P}$ is somewhere dense in ${\mathcal G}_k(\mathbb {R}^n)$.