A topological classification of convex bodies

TL;DR

This paper classifies convex bodies based on Morse-Smale complexes derived from their shape, showing the complexity of these classifications matches that of all Morse-Smale functions on the sphere through an inductive algorithm.

Contribution

It introduces an algorithm that exhaustively generates convex bodies corresponding to all combinatorial types of Morse-Smale complexes on S^2, extending previous classification schemes.

Findings

Every 2-colored quadrangulation of the sphere corresponds to a Morse-Smale complex of a convex body.

The algorithm generates convex bodies with increasingly complex Morse-Smale complexes.

The classification scheme generalizes known results and has applications to pebble shape analysis.

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 M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on 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 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 [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.