Spectrahedral representation of polar orbitopes

Abstract

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 can be described in terms of conditions on singular values and Ky Fan matrix norms.