Transversals, topology and colorful geometric results

Abstract

Suppose we have two convex sets A and B in euclidean d-space ℝd. Assume the only information we have about A and B comes from the space of their transversal lines. Can we determine whether A and B have a point in common? For example, suppose the space of their transversal lines has an essential curve; that is, suppose there is a line that moves continuously in ℝd, always remaining transversal to A and B, and comes back to itself with the opposite orientation. If this is so, then A must intersect B, otherwise there would be a hyperplane H separating A from B; but it turns out that our moving line becomes parallel to H at some point on its trip, which is a contradiction to the fact that the moving line remains transversal to the two sets. If we have three convex sets A, B and C, for example, in ℝ3, then our essential curve does not give us sufficient topological information. In this case, to detect whether A ∩ B ∩ C ≠ ϕ, we need a 2-dimensional cycle. So, for example, if we can continuously choose a transversal line parallel to every direction, then there must be a point in A ∩ B ∩ C, otherwise if not, the same is true for π(A)∩π(B)∩π(C), for a suitable orthogonal projection π: ℝ3 → H where H is a plane through the origin (see [4, Lemma 3.1]). Hence clearly there is no transversal line orthogonal to H.