Equilibria in non-Euclidean geometries

TL;DR

This paper extends the concepts of centroids and equilibrium points of convex bodies to non-Euclidean geometries like spherical, hyperbolic, and normed spaces, revealing new minimal equilibrium counts and existence of mono-monostatic bodies.

Contribution

It generalizes previous Euclidean results to non-Euclidean spaces, establishing lower bounds on equilibrium points and demonstrating the existence of mono-monostatic convex bodies.

Findings

Every plane convex body in these spaces has at least four equilibrium points.

Existence of mono-monostatic convex bodies in 3D spherical, hyperbolic, and certain normed spaces.

Generalization of Euclidean space results to non-Euclidean geometries.

Abstract

In this paper, extending the work of Gal'perin (Comm. Math. Phys. 154: 63-84, 1993), we investigate generalizations of the concepts of centroids and static equilibrium points of a convex body in spherical, hyperbolic and normed spaces. In addition, we examine the minimum number of equilibrium points a $2$- or $3$-dimensional convex body can have in these spaces. In particular, we show that every plane convex body in any of these spaces has at least four equilibrium points, and that there are mono-monostatic convex bodies in $3$-dimensional spherical, hyperbolic, and certain normed spaces. Our results are generalizations of results of Domokos, Papadopoulos and Ruina (J. Elasticity 36: 59-66, 1994), and V\'arkonyi and Domokos (J. Nonlinear Sci. 16: 255-281, 2006) for convex bodies in Euclidean space.