On the sectional area of convex polytopes

A function j : %3 -+ ‘R is unimodal if it increases to a maximum value and then decreases. It is strictly unimodal if the increase and decrease are strict. To be precise, ~ is strictly unimodal iff for all reals z < y, the value o = min{~(z), j(y)} is either the global minimum or maximum of ~ or v < ~(z) for all z G (z, y). Unimodality is important for the design of efficient search algorithms because it permits prune-and-search strategies. E.g., Chazelle and Dobkin [2] showed that the perpendicular distance from a line 1 to an n-vertex convex polygon Q is bimodal. From this, unimodal functions can be constructed and the farthest point from t can be computed in O(log n) time. Unimodality can also simplify proofs. It was shown in 1973 that if A(z) is the length of the intersection of a convex polygon Q with the vertical line through Z, then A(x) is a strictly unimodal function [4]. We note that this generalizes to higher dimensions: the area in 7?3 (or volume in 7?~) of the intersection of a convex polytope K and a hyperplane h(z) = {(x,23, ..., Zd) I VZi G ‘R} is a strictly unimodal function A(x). (If the plane is defined by rotation instead of translation, then there are convex polytopes for which sectional area is not unimodal.) Prune-andsearch can be used to compute the intersection with maximum area (volume) in time proportional to the size of K, if K is stored with its complete facial lattice. For %33 our algorithm has an application to shape matching: Given convex polygons P and Q and a direction in which to translate P, one can find the translation