The combinatorial complexity of hyperplane transversals
Sylvain E. Cappell, Jacob Eli Goodman, János Pach, Richard M. Pollack, Micha Sharir, Rephael Wenger · 1990
We show that the maximum combinatorial complexity of the space of hyperplane transversals to a family of n separated and strictly convex sets in Rd is Θ(n⌊d/2⌋), which generalizes results of Edelsbrunner and Sharir in the plane. As a key step in the argument, we show that the space of hyperplanes tangent to κ ≤ d separated and strictly convex sets in Rd is a topological (d - κ)-sphere.