Shifting Segments to Optimality

Figure 2: Near rectangular dissections in applications. Note that in the applications the areas of the rectangles of a dissection are relevant, actually in most cases these areas are prescribed. A rectangular dissection is area-universal if for any assignment of positive weights to the rectangles there is an equivalent dissection such that the areas of rectangles are equal to the given weights. The key player in this article is Theorem 2 which characterizes area-universal rectangular dissections. In Section 2 we state the theorem and discuss some proofs and generalizations. Before getting there we present two problems which do not appear to have much in common. In Section 3 we show that both problems can be solved by clever applications of the theorem.