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, including convex geometry, algebraic geometry, and optimization. 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, Carathodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, algebraic boundaries, and representations as spectrahedra or projected spectrahedra. Copyright University College London 2011.
