A topological classification of convex bodies

Domokos, Gábor and Lángi, Zsolt and Szabó, Tímea (2015) A topological classification of convex bodies. Geometriae Dedicata. ISSN 0046-5755 (Submitted)

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Abstract

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of gravity 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 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 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.

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