In this paper we rst prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let F be a collection of n totally or partially deened algebraic trivariate functions of constant maximum degree, with the additional property that, for a given pair of functions f; f 0 2 F, the surface f (x; y; z) = f 0 (x; y; z) is xy-monotone (actually, we need a somewhat weaker property|see below). We show that the vertical decomposition of the minimization diagram of F consists of O(n 3+") cells (each of constant complexity), for any " > 0. In the second part of the paper we present a general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including (i) computing the width of a point set in 3-space, (ii) computing the minimum-width annulus enclosing a set of n points in the plane, and (iii) computing thèbiggest stick' inside a simple polygon in the plane. Using the above result on vertical decompositions, we show that the expected running time of all three algorithms is O(n 3=2+"), for any " > 0. Our algorithm improves and simpliies previous solutions of all three problems.