On compacta with convex projections

In this paper it is proved that if C is a compact subset of the convex n-dimensional compactum K⊂Rn so that C and K have the same projections on all linear subspaces of some fixed dimension k for 0<k<n, then C contains a (k−1)-sphere. If K is a polyhedron then C can be chosen to be exactly (k−1)-dimensional. In the general case, C must contain some of the boundary points D of K, and C can be chosen to equal D∪Z where Z is a 0-dimensional subset of the interior of K.