A topological classification of convex bodies

摘要: The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from center mass represents a rather restricted class $${\mathcal {M}}_C$$ Morse–Smale functions on $${\mathbb {S}}^2$$ . Here we show that even exhibits complexity known for general exhausting all combinatorial possibilities: every 2-colored quadrangulation sphere is isomorphic to suitably represented complex associated function in (and vice versa). We prove our claim an inductive algorithm, starting path graph $$P_2$$ and generating corresponding quadrangulations increasing number vertices performing each combinatorially possible vertex splitting convexity-preserving local manipulation surface. Since carrying complexes exist, this algorithm not only proves but also generalizes classification scheme Varkonyi Domokos (J Nonlinear Sci 16:255–281, 2006). Our expansion essentially dual procedure presented Edelsbrunner et al. (Discrete Comput Geom 30:87–10, 2003), producing hierarchy increasingly coarse complexes. point out applications pebble shapes.