Centroids and equilibrium points of convex bodies

TL;DR

This paper surveys geometric problems involving centroids and equilibrium points of convex bodies, focusing on inequalities, classifications, and properties relevant to static equilibrium and convex geometry.

Contribution

It compiles and discusses key results on equilibrium points, inequalities, and classifications of convex bodies, highlighting recent advances and open questions.

Findings

Results related to Grünbaum's inequality and Busemann-Petty centroid inequality.

Classifications of convex bodies based on equilibrium points.

Analysis of the location, structure, and number of equilibrium points.

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 Gr\"unbaum'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.