A topological classification of convex bodies

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 $${\mathcal {M}}_C$$MC of Morse–Smale functions on $${\mathbb {S}}^2$$S2. Here we show that even $${\mathcal {M}}_C$$MC exhibits the complexity known for general Morse–Smale functions on $${\mathbb {S}}^2$$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 $${\mathcal {M}}_C$$MC (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph $$P_2$$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 $$P_2$$P2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in Várkonyi and Domokos (J Nonlinear Sci 16:255–281, 2006). Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. (Discrete Comput Geom 30:87–10, 2003), producing a hierarchy of increasingly coarse Morse–Smale complexes. We point out applications to pebble shapes.