Similarity measure computation of convex polyhedra revisited

We study the computation of rotation-invariant similarity measures of convex polyhedra, based on Minkowski’s theory of mixed volumes. To compute the similarity measure, a (mixed) volume functional has to be minimized over a number of critical orientations of these polyhedra. These critical orientations are those relative configurations where faces and edges of the two polyhedra are as much as possible parallel. Two types of critical orientations exist for two polyhedra A and B. Type-1 critical orientations are those relative orientations where a face of B is parallel to a face of A, and an edge of B is parallel to a face of A, or vice versa. Type-2 critical orientations correspond to the case that three edges of A are parallel to three faces of B, or vice versa. It has been conjectured that to perform minimization of the volume functional, it is sufficient to consider Type-1 critical orientations only. Here we present experimental proof showing this conjecture to be false.