Spectrahedral representation of polar orbitopes

Let K be a compact Lie group and V a finite-dimensional representation of K. The orbitope of a vector $$x\in V$$ is the convex hull $${\mathscr {O}}_x$$ of the orbit Kx in V. We show that if V is polar then $${\mathscr {O}}_x$$ is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope $${\mathscr {O}}_x^o$$ , which is the convex set polar to $${\mathscr {O}}_x$$ . We prove that $${\mathscr {O}}_x^o$$ is the convex hull of finitely many K-orbits, and we identify the cases in which $${\mathscr {O}}_x^o$$ is itself an orbitope. In these cases one has $${\mathscr {O}}_x^o=c\cdot {\mathscr {O}}_x$$ with $$c>0$$ . Moreover we show that if x has “rational coefficients” then $${\mathscr {O}}_x^o$$ is again a spectrahedron. This provides many new families of doubly spectrahedral orbitopes. All polar orbitopes that are derived from classical semisimple Lie algebras can be described in terms of conditions on singular values and Ky Fan matrix norms.