On compacta with convex projections (Q1296281)

Let \(C\) be a convex subset of a convex \(n\)-dimensional compactum \(K\subset\mathbb{R}^n\). Assuming that \(C\) and \(K\) have the same projections on all linear subspaces of a fixed dimension \(k\) for \(0<k<n\), the authors prove that \(C\) contains a \((k-1)\)-sphere. If \(K\) is a polyhedron then \(C\) can be chosen of dimension \(k-1\). In the general case, \(C\) must contain a subset \(D\) of boundary points of \(K\) and can be chosen to equal \(DVZ\) where \(Z\) is a \(0\)-dimensional subset in the interior of \(K\).