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Centroids and equilibrium points of convex bodies

Zsolt Lángi, Péter L. Várkonyi

TL;DR

The note surveys how centroids and static equilibrium points of convex bodies interplay with fundamental inequalities (Grünbaum's and Busemann-Petty centroid inequalities) and topology-driven classifications of equilibria. It advances a structured framework (primary, secondary, tertiary classes) to analyze how equilibria are located, counted, and transformed under deformations, including geometric constructions like Gömböc and polyhedral realizations. It also links these ideas to broader themes such as robustness of equilibrium classes, clustering phenomena, and applications to floating bodies and shape analysis. The work highlights central open questions, especially in higher-dimensional and tertiary-class transitions, and emphasizes the deep connections between convex geometry, dynamical systems, and applications in physical sciences.

Abstract

The aim of this note is to survey the results in some geometric problems related to the centroids and the static equilibrium points of convex bodies. In particular, we collect results related to Grünbaum's inequality and the Busemann-Petty centroid inequality, describe classifications of convex bodies based on equilibrium points, and investigate the location and structure of equilibrium points, their number with respect to a general reference point as well as the static equilibrium properties of convex polyhedra.

Centroids and equilibrium points of convex bodies

TL;DR

The note surveys how centroids and static equilibrium points of convex bodies interplay with fundamental inequalities (Grünbaum's and Busemann-Petty centroid inequalities) and topology-driven classifications of equilibria. It advances a structured framework (primary, secondary, tertiary classes) to analyze how equilibria are located, counted, and transformed under deformations, including geometric constructions like Gömböc and polyhedral realizations. It also links these ideas to broader themes such as robustness of equilibrium classes, clustering phenomena, and applications to floating bodies and shape analysis. The work highlights central open questions, especially in higher-dimensional and tertiary-class transitions, and emphasizes the deep connections between convex geometry, dynamical systems, and applications in physical sciences.

Abstract

The aim of this note is to survey the results in some geometric problems related to the centroids and the static equilibrium points of convex bodies. In particular, we collect results related to Grünbaum's inequality and the Busemann-Petty centroid inequality, describe classifications of convex bodies based on equilibrium points, and investigate the location and structure of equilibrium points, their number with respect to a general reference point as well as the static equilibrium properties of convex polyhedra.
Paper Structure (12 sections, 2 theorems, 34 equations, 5 figures, 1 table)

This paper contains 12 sections, 2 theorems, 34 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The centroid of a convex body
  3. Grünbaum's centroid inequality
  4. The Busemann-Petty centroid inequality
  5. Static equilibrium points
  6. Primary classes of convex bodies based on their equilibria
  7. Secondary and tertiary classes
  8. Transition between different classes
  9. The location and clustering of equilibria
  10. The number of equilibria with respect to a general reference point
  11. Equilibrium points of convex polyhedra
  12. Related problems and applications

Key Result

Theorem 2.3

Let $K \subset \mathbb R^d$ be a centered convex body. For any $u \in \mathbb{S}^{d-1}$ let $K_u$ denote the intersection of $K$ with the closed half space bounded by $u^{\perp}$ and containing $u$. Then, for any $u \in \mathbb{S}^{d-1}$, we have

Figures (5)

  • Figure 1: Equilibrium classes with some representative examples. Classes corresponding to polyhedral pairs (see Section \ref{['subsec:polyhedra']}, below) are highlighted by grey background. Unistable, uni-unstable, and mono-monostatic objects correspond to the first row, the first column, and the top-left corner of the table. The examples have been provided by Tímea Szabó and Gábor Domokos.
  • Figure 2: Left: illustration of the Gömböc: a piecewise smooth body in class $\{1,1\}$. The boundary is coloured according to level curves of the radial function, which has two critical points at the top and at the bottom of the surface. Right: the monostatic polyhedron of Guy conway1966stability, see Section \ref{['subsec:polyhedra']}, below. The illustration has been created with the aid of a 3D model available at https://www.thingiverse.com/thing:90866.
  • Figure 3: The topological graphs of convex bodies with at most two stable and two unstable points.
  • Figure 4: Different representations of a Morse-Smale complex. a) Primal Morse-Smale graph. b) Primal Morse-Smale graph with the stable and unstable points connected. c) Quasi-dual Morse-Smale graph. d) Quasi-dual graph with the stable points connected. e) The induced uncolored planar graph; vertices, edges and faces correspond to the stable, saddle and unstable points of the primal representation.
  • Figure 5: Two examples of smooth, planar curves with their evolutes, and the corresponding number of equilibria. In the left panel, the dashed circle illustrates the internal robustness of a point P. The right one is an example with $n_{min,K}=4$.

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Definition 3.5
  • Definition 3.6