More on compacta with convex projections. (Q1852367)

The authors deal with the phenomenon of nonconvex compact sets having convex projections (shadows) onto every hyperplane. They establish some properties of such sets and prove existence of such a set even with a prescribed condition. The main results are the following 2 theorems. Theorem 1. If \(C\) is a nonconvex compact set in the Euclidean \(n\)-space, \(n>2\), with all its shadows convex, then either (1) \(C\) contains an \((n-1)\)-cell (i.e. a subset homeomorphic to the standard closed \((n-1)\)-dimensional disc), or (2) there exist \(n+1\) distinct hyperplanes such that the intersection of \(C\) with any of them contains a topological \((n-2)\)-sphere. Theorem 4. For every \(n>2\), there exists a nonconvex compact set \(C\) in the Euclidean \(n\)-space with all its shadows convex such that there are precisely three distinct hyperplanes whose intersection with \(C\) contains a topological \((n-2)\)-sphere. (Application of Theorem 1 to the set \(C\) from Theorem 4 shows that \(C\) must contain an \((n-1)\)-cell.)