Topological transversals to a family of convex sets
Let F be a family of compact convex sets in Rd. We say that F has a topological rho-transversal of index (m, k) (rho < m, 0 < k <= d -m) if there are, homologically, as many transversal m-planes to F as m-planes containing a fixed rho-plane in Rm+k. Clearly, if F has a rho-transversal plane, then F has a topological rho-transversal of index (m, k), for rho < m and k <= d - m. The converse is not true in general. We prove that for a family F of rho + k + 1 compact convex sets in R-d a topological rho-transversal of index (m, k) implies an ordinary rho-transversal. We use this result, together with themultiplication formulas for Schubert cocycles, the LusternikSchnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by Barany and Lovasz, to obtain some geometric consequences.
common transversal the helly theorem the schubert calculus