Common tangents and common transversals (Q1333230)

A family of compact convex sets in \(\mathbb{R}^ d\) is called separated if no \(k\leq d\) of them are simultaneously met by any \((k-2)\)-dimensional affine linear subspace. In the first part of the paper, the authors prove that the set of oriented common tangent hyperplanes to a given separated family of \(k\leq d\) compact convex sets in \(\mathbb{R}^ d\) is homeomorphic to the sphere \(S^{d-k}\). The proof uses the topological Poincaré conjecture in dimensions greater than 3. For dimension 3 there is a direct argument. This raises the question whether a more geometric proof is possible without referring to deep results on topological manifolds by Kriby- Siebenmann and Freedman. In the second part the complexity of the space of transversals is studied. More precisely, for any separated family \(A\) of \(n\) compact strictly convex sets in \(\mathbb{R}^ d\) let \(\tau (A)\) denote the space of all hyperplanes meeting every member of \(A\). Then the number of connected components of \(\tau (A)\) is \(O (n^{d-1})\).