A topological classification of convex bodies

G. Domokos, Zsolt Lángi, Tímea Szabó

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish
Pages (from-to)1-22
Number of pages22
JournalGeometriae Dedicata
DOIs
Publication statusAccepted/In press - Nov 27 2015

Fingerprint

Convex Body
Quadrangulation
Isomorphic
Euclidean Distance
Barycentre
Convexity
Manipulation
Critical point
Path
Generalise
Graph in graph theory
Vertex of a graph

Keywords

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

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

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

In: Geometriae Dedicata, 27.11.2015, p. 1-22.

Research output: Contribution to journalArticle

Domokos, G. ; Lángi, Zsolt ; Szabó, Tímea. / A topological classification of convex bodies. In: Geometriae Dedicata. 2015 ; pp. 1-22.
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