A characterization of the symmetry groups of mono-monostatic convex bodies

TL;DR

This paper characterizes the symmetry groups of mono-monostatic convex bodies, extending previous results by providing a comprehensive classification for both smooth convex bodies and polyhedra, thus answering longstanding questions in geometric symmetry.

Contribution

It offers a complete characterization of the symmetry groups possible for mono-monostatic convex bodies and polyhedra, advancing understanding of their geometric and symmetry properties.

Findings

Characterization of symmetry groups for smooth convex bodies

Characterization of symmetry groups for convex polyhedra

Answers to longstanding symmetry questions in convex geometry

Abstract

Answering a question of Conway and Guy in a 1968 paper, L\'angi in 2021 proved the existence of a monostable polyhedron with $n$-fold rotational symmetry for any $n \geq 3$, and arbitrarily close to a Euclidean ball. In this paper we strengthen this result by characterizing the possible symmetry groups of all mono-monostatic smooth convex bodies and convex polyhedra. Our result also answers a stronger version of the question of Conway and Guy, asked in the above paper of L\'angi.