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

Abstract

This paper shows that the i-level of an arrangement of hyperplanes in Ed has at most {Mathematical expression} 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 in Ed with n facets. © 1993 Springer-Verlag New York Inc.