Orbitopes

TL;DR

This paper develops a comprehensive theory of orbitopes, convex bodies formed from group orbits, focusing on their geometric, algebraic, and optimization properties, especially for groups SO(n) and O(n).

Contribution

It provides a unified, self-contained framework for understanding orbitopes, including their structure, boundary, and spectrahedral representations, with detailed analysis of specific classes.

Findings

Characterization of face lattices and boundary hypersurfaces.

Representation of orbitopes as spectrahedra or projected spectrahedra.

Analysis of orbitopes arising from SO(n) and O(n) groups.

Abstract

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, in particular convex geometry, optimization, and algebraic geometry. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n). These include Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese orbitopes and Grassmann orbitopes. We study their face lattices, their algebraic boundary hypersurfaces, and representations as spectrahedra or projected spectrahedra.