A bound on local minima of arrangements that implies the upper bound theorem

This paper shows that thei-level of an arrangement of hyperplanes inE ^d has at most local minima. This bound follows from ideas previously used to prove bounds on (≤k)-sets. Using linear programming duality, the Upper Bound Theorem is obtained as a corollary, giving yet another proof of this celebrated bound on the number of vertices of a simple polytope inE ^d withn facets.