A topological classification of convex bodies

Abstract

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 ${\mathcal {M}}_C$ of Morse–Smale functions on ${\mathbb {S}}^2$. Here we show that even ${\mathcal {M}}_C$ exhibits the complexity known for general Morse–Smale functions on ${\mathbb {S}}^2$ 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 ${\mathcal {M}}_C$ (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph $P_2$ 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 $P_2$ 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.