This paper provides an alternative viewpoint of multidimensional bisection global optimisation methods of Wood. A dual coordinate representation of convex bodies is introduced which leads to an easy implementation and eliminates the need to see the geometry of intersecting simplexes. Although developed in the context of global optimisation, the techniques deal more generally with regions represented as the union of convex bodies. With this dual framework the algorithm can be implemented efficiently using any multi-attribute index data structure that allows for quick range queries. A C version using a "multi-key double linked skip list" based on Pugh's skip list has been implemented.