### 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 (Formula presented.) of Morse–Smale functions on (Formula presented.). Here we show that even (Formula presented.) exhibits the complexity known for general Morse–Smale functions on (Formula presented.) 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 (Formula presented.) (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph (Formula presented.) 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 (Formula presented.) 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.

Original language | English |
---|---|

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Geometriae Dedicata |

DOIs | |

Publication status | Accepted/In press - Nov 27 2015 |

### Fingerprint

### Keywords

- Convex surface
- Equilibrium
- Morse–Smale complex
- Pebble shape
- Quadrangulation
- Vertex splitting

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometriae Dedicata*, 1-22. https://doi.org/10.1007/s10711-015-0130-4

**A topological classification of convex bodies.** / Domokos, G.; Lángi, Zsolt; Szabó, Tímea.

Research output: Contribution to journal › Article

*Geometriae Dedicata*, pp. 1-22. https://doi.org/10.1007/s10711-015-0130-4

}

TY - JOUR

T1 - A topological classification of convex bodies

AU - Domokos, G.

AU - Lángi, Zsolt

AU - Szabó, Tímea

PY - 2015/11/27

Y1 - 2015/11/27

N2 - 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 (Formula presented.) of Morse–Smale functions on (Formula presented.). Here we show that even (Formula presented.) exhibits the complexity known for general Morse–Smale functions on (Formula presented.) 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 (Formula presented.) (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph (Formula presented.) 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 (Formula presented.) 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.

AB - 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 (Formula presented.) of Morse–Smale functions on (Formula presented.). Here we show that even (Formula presented.) exhibits the complexity known for general Morse–Smale functions on (Formula presented.) 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 (Formula presented.) (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph (Formula presented.) 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 (Formula presented.) 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.

KW - Convex surface

KW - Equilibrium

KW - Morse–Smale complex

KW - Pebble shape

KW - Quadrangulation

KW - Vertex splitting

UR - http://www.scopus.com/inward/record.url?scp=84948671357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948671357&partnerID=8YFLogxK

U2 - 10.1007/s10711-015-0130-4

DO - 10.1007/s10711-015-0130-4

M3 - Article

AN - SCOPUS:84948671357

SP - 1

EP - 22

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

ER -