Bounding the Number of Geometric Permutations Induced byk-Transversals

We prove that a suitably separated family ofncompact convex sets inRdcan be met byk-flat transversals in at mostO(k)d2<//SUP> ((2k+1−2<//SUP>k)(nk+1))k(d−k), or for fixedkandd,O(nk(k+1)(d−k)) different order types. This is the first non-trivial upper bound for 1<k<d−1, and generalizes (asymptotically) the best upper bounds known for line transversals inR,d>2.