A topological classification of convex bodies

الأصل · EN

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class MC of Morse-Smale functions on S². Here we show that even MC exhibits the complexity known for general Morse-Smale functions on S² by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in MC (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P₂ and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P₂ exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.